Loading [MathJax]/jax/element/mml/optable/BasicLatin.js
Research article Special Issues

Global existence for reaction-diffusion evolution equations driven by the p-Laplacian on manifolds

  • We consider reaction-diffusion equations driven by the p-Laplacian on noncompact, infinite volume manifolds assumed to support the Sobolev inequality and, in some cases, to have L2 spectrum bounded away from zero, the main example we have in mind being the hyperbolic space of any dimension. It is shown that, under appropriate conditions on the parameters involved and smallness conditions on the initial data, global in time solutions exist and suitable smoothing effects, namely explicit bounds on the L norm of solutions at all positive times, in terms of Lq norms of the data. The geometric setting discussed here requires significant modifications w.r.t. the Euclidean strategies.

    Citation: Gabriele Grillo, Giulia Meglioli, Fabio Punzo. Global existence for reaction-diffusion evolution equations driven by the p-Laplacian on manifolds[J]. Mathematics in Engineering, 2023, 5(3): 1-38. doi: 10.3934/mine.2023070

    Related Papers:

    [1] La-Su Mai, Suriguga . Local well-posedness of 1D degenerate drift diffusion equation. Mathematics in Engineering, 2024, 6(1): 155-172. doi: 10.3934/mine.2024007
    [2] Giacomo Ascione, Daniele Castorina, Giovanni Catino, Carlo Mantegazza . A matrix Harnack inequality for semilinear heat equations. Mathematics in Engineering, 2023, 5(1): 1-15. doi: 10.3934/mine.2023003
    [3] David Cruz-Uribe, Michael Penrod, Scott Rodney . Poincaré inequalities and Neumann problems for the variable exponent setting. Mathematics in Engineering, 2022, 4(5): 1-22. doi: 10.3934/mine.2022036
    [4] Pawan Kumar, Christina Surulescu, Anna Zhigun . Multiphase modelling of glioma pseudopalisading under acidosis. Mathematics in Engineering, 2022, 4(6): 1-28. doi: 10.3934/mine.2022049
    [5] Antonio Iannizzotto, Giovanni Porru . Optimization problems in rearrangement classes for fractional p-Laplacian equations. Mathematics in Engineering, 2025, 7(1): 13-34. doi: 10.3934/mine.2025002
    [6] Daniele Castorina, Giovanni Catino, Carlo Mantegazza . A triviality result for semilinear parabolic equations. Mathematics in Engineering, 2022, 4(1): 1-15. doi: 10.3934/mine.2022002
    [7] Anne-Charline Chalmin, Jean-Michel Roquejoffre . Improved bounds for reaction-diffusion propagation driven by a line of nonlocal diffusion. Mathematics in Engineering, 2021, 3(1): 1-16. doi: 10.3934/mine.2021006
    [8] Giuseppe Maria Coclite, Lorenzo di Ruvo . On the initial-boundary value problem for a Kuramoto-Sinelshchikov type equation. Mathematics in Engineering, 2021, 3(4): 1-43. doi: 10.3934/mine.2021036
    [9] Raúl Ferreira, Arturo de Pablo . A nonlinear diffusion equation with reaction localized in the half-line. Mathematics in Engineering, 2022, 4(3): 1-24. doi: 10.3934/mine.2022024
    [10] Antonio Vitolo . Singular elliptic equations with directional diffusion. Mathematics in Engineering, 2021, 3(3): 1-16. doi: 10.3934/mine.2021027
  • We consider reaction-diffusion equations driven by the p-Laplacian on noncompact, infinite volume manifolds assumed to support the Sobolev inequality and, in some cases, to have L2 spectrum bounded away from zero, the main example we have in mind being the hyperbolic space of any dimension. It is shown that, under appropriate conditions on the parameters involved and smallness conditions on the initial data, global in time solutions exist and suitable smoothing effects, namely explicit bounds on the L norm of solutions at all positive times, in terms of Lq norms of the data. The geometric setting discussed here requires significant modifications w.r.t. the Euclidean strategies.



    We investigate existence of nonnegative global in time solutions to the quasilinear parabolic problem

    {ut=div(|u|p2u)+uσinM×(0,T)u=u0inM×{0}, (1.1)

    where M is an N-dimensional, complete, noncompact, Riemannian manifold of infinite volume, whose metric is indicated by g, and where div and are respectively the divergence and the gradient with respect to g and T(0,+]. We shall assume throughout this paper that

    2NN+1<p<N,σ>p1. (1.2)

    The problem is posed in the Lebesgue spaces

    Lq(M)={v:MRmeasurable,vLq:=(Mvqdμ)1/q<+},

    where μ is the Riemannian measure on M. We also assume the validity of the Sobolev inequality:

    (Sobolev inequality)vLp(M)1Cs,pvLp(M)for anyvCc(M), (1.3)

    where Cs,p>0 is a constant and p:=pNNp. In some cases we also assume that the Poincaré inequality is valid, that is

    (Poincaré inequality)vLp(M)1CpvLp(M)for anyvCc(M), (1.4)

    for some Cp>0. Observe that, for instance, (1.3) holds if M is a Cartan-Hadamard manifold, i.e., a simply connected Riemannian manifold with nonpositive sectional curvatures, while (1.4) is valid when M is a Cartan-Hadamard manifold satisfying the additional condition of having sectional curvatures bounded above by a constant c<0 (see, e.g., [15,16]). Therefore, as it is well known, on RN (1.3) holds, but (1.4) fails, whereas on the hyperbolic space both (1.3) and (1.4) are fulfilled.

    Global existence and finite time blow-up of solutions for problem (1.1) has been deeply studied when M=RN, especially in the case p=2 (linear diffusion). The literature for this problem is huge and there is no hope to give a comprehensive review here. We just mention the fundamental result of Fujita, see [10], who shows that blow-up in a finite time occurs for all nontrivial nonnegative data when σ<1+2N, while global existence holds, for σ>1+2N, provided the initial datum is small enough in a suitable sense. Furthermore, the critical exponent σ=1+2N, belongs to the case of finite time blow-up, see e.g., [22] for the one dimensional case, N=1, or [23] for N>1. For further results concerning problem (1.1) with p=2 see e.g., [7,9,11,20,26,34,35,36,41,42,43]).

    Similarly, the case of problem (1.1) when M=RN and p>1 has attracted a lot of attention, see e.g., [12,13,14,30,31,32,33] and references therein. In particular, in [31], nonexistence of nontrivial weak solutions is proved for problem (1.1) with M=RN and

    p>2NN+1,max

    Similar weighted problems have also been treated. In fact, for any strictly positive measurable function \rho: {\mathbb{R}}^N\to {\mathbb{R}} , let us consider the weighted { \rm L}^{q}_{\rho} spaces

    \begin{align} { \rm L}^q_{\rho}( {\mathbb{R}}^N) = \left\{v: {\mathbb{R}}^N\to {\mathbb{R}}\,\, {\text{measurable}}\,\, , \,\, \|v\|_{L^q_{\rho}}: = \left(\int_{ {\mathbb{R}}^N} v^q\rho(x)\,dx\right)^{1/q} < +\infty\right\}. \end{align}

    In [27] problem

    \begin{equation} \begin{cases} \rho(x) u_t = \operatorname{div} \left(|\nabla u|^{p-2}\nabla u \right)+\, \rho(x)u^{\sigma} & {\text{in}}\,\, {\mathbb{R}}^N \times (0,T) \\ u = u_0 &{\text{in}}\,\, {\mathbb{R}}^N\times \{0\}\,, \end{cases} \end{equation} (1.5)

    is addressed. In [27,Theorem 1], it is showed that, when p > 2 , \rho(x) = (1+|x|)^{-l} , 0\le l < p , \sigma > p-1+\frac pN , u_0\in { \rm L}^1_{\rho}({\mathbb{R}}^N)\cap { \rm L}^s_{\rho}({\mathbb{R}}^N) is sufficiently small, with s > \frac{(N-l)(\sigma-p+1)}{p-l} , then problem (1.5) admits a global in time solution. Moreover, the solution satisfies a smoothing estimate { \rm L}^1_{\rho}-{ \rm L}^{\infty} , in the sense that for sufficiently small data u_0\in { \rm L}^1_{\rho}({\mathbb{R}}^N) , the corresponding solution is bounded, and a quantitive bound on the { \rm L}^{\infty} norm of the solution holds, in term of the { \rm L}^1_{\rho}({\mathbb{R}}^N) norm of the initial datum. On the other hand, in [27,Theorem 2], when p > 2 , \rho(x) = (1+|x|)^{-l} , l\ge p , \sigma > p-1 , u_0\in { \rm L}^1_{\rho}({\mathbb{R}}^N)\cap { \rm L}^s_{\rho}({\mathbb{R}}^N) is sufficiently small, with s > \frac{N}{p}(\sigma-p+1) , then problem (1.5) admits a global in time solution, which is bounded for positive times.

    On the other hand, existence and nonexistence of global in time solutions to problems closely related to problem (1.1) have been investigated also in the Riemannian setting. The situation can be significantly different from the Euclidean situation, especially in the case of negative curvature. Infact, when dealing with the case of the N -dimensional hyperbolic space, M = \mathbb{H}^N , it is known that when p = 2 , for all \sigma > 1 and sufficiently small nonnegative data there exists a global in time solution, see [3,34,39,40]. A similar result has been also obtain when M is a complete, noncompact, stochastically complete Riemannian manifolds with \lambda_1(M) > 0 , where \lambda_1(M): = \inf \operatorname{spec}(-\Delta) , see [19]. Stochastic completeness amounts to requiring that the linear heat semigroup preserves the identity, and is known to hold e.g., if the sectional curvature satisfies {\text{sec}}(x)\, \ge -c d(x, o)^2 for all x\in M outside a given compact, and a suitable c > 0 , where d is the Riemannian distance and o is a given pole. Besides, it is well known that \lambda_1(M) > 0 e.g., if {\text{sec}}(x)\, \le -c < 0 for all x\in M . Therefore, the class of manifolds for which the results of [19] hold is large, since it includes e.g., all Cartan-Hadamard manifolds with curvature bounded away from zero and not diverging faster than quadratically at infinity.

    Concerning problem (1.1) with p > 1 , we refer the reader to [28,29] and references therein. In particular, in [28], nonexistence of global in time solutions on infinite volume Riemannian manifolds M is shown under suitable weighted volume growth conditions. In [29], problem (1.1) with M = \Omega being a bounded domain and u^{\sigma} replaced by V(x, t)u^{\sigma} is addressed, where V is a positive potential. To be specific, nonexistence of nonnegative, global solutions is established under suitable integral conditions involving V , p and \sigma .

    In this paper, we prove the following results. Assume that the bounds (1.2) and the Sobolev inequality (1.3) hold, and besides that \sigma > p-1+\frac pN .

    (a) If u_0\in { \rm L}^{s}(M)\cap { \rm L}^1(M) is sufficiently small, with s > (\sigma-p+1)\frac Np , then a global solution exists. Furthermore, a smoothing estimate of the type { \rm L}^1-{ \rm L}^{\infty} holds (see Theorem 2.2).

    (b) If u_0\in { \rm L}^{(\sigma-p+1)\frac Np}(M) is sufficiently small, then a global solution exists. Furthermore, a smoothing estimate of the type { \rm L}^{(\sigma-p+1)\frac Np}-{ \rm L}^{\infty} holds (see Theorem 2.4), this being new even in the Euclidean case.

    (c) In addition, in both the latter two cases, we establish a { \rm L}^{(\sigma-p+1)\frac Np}-{ \rm L}^q smoothing estimate, for any (\sigma-p+1)\frac Np\le q < +\infty and an { \rm L}^q-{ \rm L}^q estimate for any 1 < q < +\infty , for suitable initial data u_0 .

    Now suppose that both the Sobolev inequality (1.3) and the Poincaré inequality (1.4) hold, and that (1.2) holds. This situation has of course no Euclidean analogue, as it is completely different from the case of a bounded Euclidean domain since M is noncompact and of infinite measure. Then:

    (d) If u_0\in { \rm L}^{s}(M)\cap { \rm L}^{\sigma\frac Np}(M) is sufficiently small, with s > \max\left\{(\sigma-p+1)\frac Np, 1\right\} , then a global solution exists. Furthermore, a smoothing estimate of the type { \rm L}^s-{ \rm L}^{\infty} holds (see Theorem 2.7).

    (e) In addition, we establish and { \rm L}^{\sigma\frac Np}-{ \rm L}^q estimate, for any \sigma\frac Np\le q < +\infty and an { \rm L}^q-{ \rm L}^q estimate for any 1 < q < +\infty , for suitable initial data u_0 .

    Note that, when we require both (1.3) and (1.4), the assumption on \sigma can be relaxed.

    In order to prove (a), we adapt the methods exploited in [27,Theorem 1]. Moreover, (b), (c) and (e) are obtained by means of an appropriate use of the Moser iteration technique, see also [18] for a similar result in the case of the porous medium equation with reaction. The proof of statement (d) is inspired [27,Theorem 2]; however, significant changes are needed since in [27] the precise form of the weight \rho is used.

    As concerns smoothing effects for general nonlinear evolution equations, we refer the reader to the fundamental works of Bénilan [4] and, slightly later but with considerable further generality and methodological simplifications, Véron [38]. Recently, Coulhon and Hauer further generalize such results and give new and abstract ones which even allow to avoid Moser's iteration in a very general functional analytic setting, through an extrapolation argument, see [8]. It should also be remarked that, though we deal with weak solutions to our problems, it is certainly possible to prove existence of solution in stronger senses, e.g., the strong one according to Bénilan and Crandall seminal contribution [5]. In this regard, we also refer to the recent paper [21], in which existence results are proved also for parabolic equations governed by the p-Laplace operator with Lipschitz lower-order terms. We also mention that several important and seminal contributions to regularity results for solutions of general nonlinear parabolic equations and systems can be found in several works by Mingione, see e.g., [1,6,24].

    The paper is organized as follows. The main results are stated in Section 2. Section 3 is devoted to { \rm L}^{q_0}-{ \rm L}^q and { \rm L}^q-{ \rm L}^q smoothing estimates, mainly instrumental to what follows. Some a priori estimates are obtained in Section 4. In Sections 5–7, Theorems 2.2, 2.4 and 2.7 are proved, respectively. Finally, in Section 8 we state similar results for the porous medium equation with reaction; the proofs are omitted since they are entirely similar to the p-Laplacian case.

    Solutions to (1.1) will be meant in the weak sense, according to the following definition.

    Definition 2.1. Let M be a complete noncompact Riemannian manifold of infinite volume. Let p > 1 , \sigma > p-1 and u_0\in{ \rm L}^{1}_{loc}(M) , u_0\ge0 . We say that the function u is a weak solution to problem (1.1) in the time interval [0, T) if

    u\in { \rm L}^2((0,T);{ \rm W}^{1,p}_{loc}(M)) \cap { \rm L}^{\sigma}_{loc}(M\times(0,T))

    and for any \varphi \in C_c^{\infty}(M\times[0, T]) such that \varphi(x, T) = 0 for any x\in M , u satisfies the equality:

    \begin{equation*} \begin{aligned} -\int_0^T\int_{M} \,u\,\varphi_t\,d\mu\,dt = &-\int_0^T\int_{M} |\nabla u|^{p-2}\left \langle \nabla u, \nabla \varphi \right \rangle\,d\mu\,dt\,+ \int_0^T\int_{M} \,u^{\sigma}\,\varphi\,d\mu\,dt \\ & +\int_{M} \,u_0(x)\,\varphi(x,0)\,d\mu. \end{aligned} \end{equation*}

    First we consider the case that \sigma > p-1+\frac p N and that the Sobolev inequality holds on M . In order to state our results, we define

    \begin{equation} \sigma_0: = (\sigma-p+1)\frac{N}{p}. \end{equation} (2.1)

    Observe that \sigma_0 > 1 whenever \sigma > p-1+\frac pN . Our first result is a generalization of [27] to the geometric setting considered here.

    Theorem 2.2. Let M be a complete, noncompact, Riemannian manifold of infinite volume such that the Sobolev inequality (1.3) holds. Assume (1.2) holds and, besides, that \sigma > p-1+\frac pN , s > \sigma_0 and u_0\in{ \rm L}^{s}(M)\cap L^1(M) , u_0\ge0 where \sigma_0 has been defined in (2.1).

    (ⅰ) Assume that

    \begin{equation} \|u_0\|_{ \rm L^{s}(M)}\, < \,\varepsilon_0,\quad \|u_0\|_{ \rm L^{1}(M)} < \,\varepsilon_0\,, \end{equation} (2.2)

    with \varepsilon_0 = \varepsilon_0(\sigma, p, N, C_{s, p}) > 0 sufficiently small. Then problem (1.1) admits a solution for any T > 0 , in the sense of Definition 2.1. Moreover, for any \tau > 0, one has u\in L^{\infty}(M\times(\tau, +\infty)) and there exists a constant \Gamma > 0 such that, one has

    \begin{equation} \|u(t)\|_{L^{\infty}(M)}\le \Gamma\, t^{-\alpha}\,\|u_0\|_{L^{1}(M)}^{\frac{p}{N(p-2)+p}}\,\quad \mathit{{\text{for all t > 0 ,}}} \end{equation} (2.3)

    where

    \begin{equation*} \alpha: = \frac{N}{N(p-2)+p}\,. \end{equation*}

    (ⅱ) Let \sigma_0\le q < \infty . If

    \begin{equation} \|u_0\|_{L^{\sigma_0}(M)} < \hat \varepsilon_0 \end{equation} (2.4)

    for \hat\varepsilon_0 = \hat\varepsilon_0(\sigma, p, N, C_{s, p}, q) > 0 small enough, then there exists a constant C = C(\sigma, p, N, \hat\varepsilon_0, C_{s, p}, q) > 0 such that

    \begin{equation} \|u(t)\|_{L^q(M)}\le C\,t^{-\gamma_q} \|u_{0}\|^{\delta_q}_{L^{\sigma_0}(M)}\quad for\; all\,\, t > 0\,, \end{equation} (2.5)

    where

    \gamma_q = \frac{1}{\sigma-1}\left[1-\frac{N(\sigma-p+1)}{pq}\right],\quad \delta_q = \frac{\sigma-p+1}{\sigma-1}\left[1+\frac{N(p-2)}{pq}\right]\,.

    (ⅲ) Finally, for any 1 < q < \infty , if u_0\in { \rm L}^q(M)\cap \rm L^{\sigma_0}(M) and

    \begin{equation} \|u_0\|_{ \rm L^{\sigma_0}(M)}\, < \,\varepsilon \end{equation} (2.6)

    with \varepsilon = \varepsilon(\sigma, p, N, C_{s, p}, q) > 0 sufficiently small, then

    \begin{equation} \|u(t)\|_{L^q(M)}\le \|u_{0}\|_{L^q(M)}\quad for\; all\,\, t > 0\,. \end{equation} (2.7)

    Remark 2.3. Observe that the choice of \varepsilon_0 in (2.2) is made in Lemma 5.1. Moreover, the proof of the above theorem will show that one can take an explicit value of \hat \varepsilon_0 in (2.4) and \varepsilon in (2.6). In fact, let q_0 > 1 be fixed and \{q_n\}_{n\in\mathbb{N}} be the sequence defined by:

    q_n = \frac{N}{N-p}(p+q_{n-1}-2), \quad {\text{for all}}\,\,n\in\mathbb{N},

    so that

    \begin{equation} q_n = \left(\frac{N}{N-p}\right)^{n}q_0+\frac{N}{N-p}(p-2) \sum\limits_{i = 0}^{n-1} \left(\frac{N}{N-p}\right)^i. \end{equation} (2.8)

    Clearly, \{q_n\} is increasing and q_n \longrightarrow +\infty as n\to +\infty. Fix q\in[q_0, +\infty) and let \bar n be the first index such that q_{\bar n}\ge q . Define

    \begin{equation} \tilde\varepsilon_0 = \tilde\varepsilon_0(\sigma,p,N,C_{s,p},q,q_0): = \left[\min \left\{\min\limits_{n = 0,...,\bar n}\left(\frac{p( q_n-1)^{1/p}}{p+q_n-2}\right)^p;\left(\frac{p(\sigma_0-1)^{1/p}}{p-\sigma_0-2}\right)^p\right\}\frac{C_{s,p}^p}{2}\right]^{\frac{1}{\sigma-p+1}}. \end{equation} (2.9)

    Observe that \tilde\varepsilon_0 in (2.9) depends on the value q through the sequence \{q_n\} . More precisely, \bar n is increasing with respect to q , while the quantity \min_{n = 0, ..., \bar n}(q_n-1)\left(\frac{p}{p+q_n-2}\right)^p\frac{C_{s, p}^p}{2} decreases w.r.t. q .

    Then, in (2.4) we can take

    \begin{equation*} \label{epsilon0aaa} \hat \varepsilon_0 = \hat\varepsilon_0(\sigma, p, N, C_{s,p}, q) = \tilde\varepsilon_0(\sigma, p, N, C_{s,p}, q, \sigma_0)\,. \end{equation*}

    Similarly, in (2.6), we can take

    \begin{equation*} \varepsilon = \bar\varepsilon_0\wedge \hat\varepsilon_0, \end{equation*}

    where

    \bar\varepsilon_0 = \bar\varepsilon_0(\sigma, p, C_{s,p}, q): = \left[\min \left\{\left(\frac{p( q-1)^{1/p}}{p+q-2}\right)^pC_{s,p}^p;\,\,\left(\frac{p(\sigma_0-1)^{1/p}}{p-\sigma_0-2}\right)^pC_{s,p}^p\right\}\right]^{\frac1{\sigma-p+1}}\,.

    The next result involves a similar smoothing effect for a different class of data. Such result seems to be new also in the Euclidean setting.

    Theorem 2.4. Let M be a complete, noncompact, Riemannian manifold of infinite volume such that the Sobolev inequality (1.3) holds. Assume (1.2) and, besides, that \sigma > p-1+\frac pN and u_0\in{ \rm L}^{\sigma_0}(M) , u_0\ge0 , with \sigma_0 as in (2.1). Assume that

    \begin{equation} \|u_0\|_{ \rm L^{\sigma_0}(M)}\, < \,\varepsilon_2\,, \end{equation} (2.10)

    with \varepsilon_2 = \varepsilon_2(\sigma, p, N, C_{s, p}, q) > 0 sufficiently small. Then problem (1.1) admits a solution for any T > 0 , in the sense of Definition 2.1. Moreover, for any \tau > 0, one has u\in L^{\infty}(M\times(\tau, +\infty)) and for any \sigma > \sigma_0 , there exists a constant \Gamma > 0 such that, one has

    \begin{equation} \|u(t)\|_{L^{\infty}(M)}\le \Gamma\, t^{-\frac1{\sigma-1}}\,\|u_0\|_{L^{\sigma_0}(M)}^{\frac{\sigma-p+1}{\sigma-1}}\, \quad \mathit{{\text{for all $t > 0$.}}} \end{equation} (2.11)

    Moreover, (ⅱ) and (ⅲ) of Theorem 2.2 hold.

    Remark 2.5. We comment that, as in Remark 2.3, one can choose an explicit value for \varepsilon_2 in (2.10). In fact, let q_0 = \sigma_0 in (2.9). It can be shown that one can take, with this choice of q_0 :

    \varepsilon_2 = \varepsilon_2(\sigma,p,N, C_{s,p},\sigma_0): = \min\left\{\tilde \varepsilon_0(\sigma, p, N, C_{s,p}, q, \sigma_0)\,;\left(\frac{1}{C\,\tilde C}\right)^{\frac 1{\sigma-p+1}}\right\}\,,

    where C > 0 and \tilde C > 0 are defined in Proposition 3.3 and Lemma 4.3, respectively.

    Remark 2.6. Observe that, due to the assumption \sigma > p-1+\frac pN , one has

    \frac 1{\sigma-1} < \frac{N}{N(p-2)+p}.

    Hence, for large times, the decay given by Theorem 2.4 is worse than the one of Theorem 2.2; however, in this regards, note that the assumptions on the initial datum u_0 are different in the two theorems. On the other hand, estimates (2.11) and (2.3), are not sharp in general for small times. For example, when u_0\in L^{\infty}(M) , u(t) remains bounded for any t\in[0, T) , where T is the maximal existence time.

    In the next theorem, we address the case \sigma > p-1 , assuming that both the inequalities (1.3) and (1.4) hold on M , hence with stronger assumptions on the manifold considered. This has of course no Euclidean analogue, as the noncompactness of the manifold considered, as well as the fact that it has infinite volume, makes the situation not comparable to the case of a bounded Euclidean domain.

    Theorem 2.7. Let M be a complete, noncompact manifold of infinite volume such that the Sobolev inequality (1.3) and the Poincaré inequality (1.4) hold. Assume that (1.2) holds, and besides that p > 2 . Let u_0\ge0 be such that u_0\in{ \rm L}^{s}(M)\cap { \rm L}^{\sigma\frac Np}(M) , for some s > \max\left\{\sigma_0, 1\right\} and q_0 > 1 . Assume also that

    \begin{equation*} \label{epsilon1} \|u_0\|_{ \rm L^{s}(M)}\, < \,\varepsilon_1, \quad \|u_0\|_{ \rm L^{\sigma\frac Np}(M)}\, < \,\varepsilon_1, \end{equation*}

    with \varepsilon_1 = \varepsilon_1(\sigma, p, N, C_{s, p}, C_p, s) sufficiently small. Then problem (1.1) admits a solution for any T > 0 , in the sense of Definition 2.1. Moreover, for any \tau > 0, one has u\in L^{\infty}(M\times(\tau, +\infty)) and, for any q > s , there exists a constant \Gamma > 0 such that, one has

    \begin{equation} \|u(t)\|_{L^{\infty}(B_R)}\le \Gamma\, t^{-\beta_{q,s}}\,\|u_0\|_{L^{s}(B_R)}^{\frac{ps}{N(p-2)+pq}}\,\quad for \;all\;\;\;t > 0 , \end{equation} (2.12)

    where

    \begin{equation*} \label{beta} \beta_{q,s}: = \frac{1}{p-2}\left(1-\frac{ps}{N(p-2)+pq}\right) > 0\,. \end{equation*}

    Moreover, let s\le q < \infty and

    \begin{equation*} \label{eps3b} \|u_0\|_{L^{s}(M)} < \hat \varepsilon_1 \end{equation*}

    for \hat\varepsilon_1 = \hat\varepsilon_1(\sigma, p, N, C_{s, p}, C_p, q, s) small enough. Then there exists a constant C = C(\sigma, p, N, \varepsilon_1, C_{s, p}, C_p, q, s) > 0 such that

    \begin{equation} \|u(t)\|_{L^q(M)}\le C\,t^{-\gamma_q} \|u_{0}\|^{\delta_q}_{L^{s}(M)}\quad for\; all \,\, t > 0\,, \end{equation} (2.13)

    where

    \gamma_q = \frac{s}{p-2}\left[\frac 1s-\frac 1q\right],\quad \delta_q = \frac sq\,.

    Finally, for any 1 < q < \infty , if u_0\in { \rm L}^q(M)\cap \rm L^{s}(M) and

    \begin{equation*} \label{eps2b} \|u_0\|_{ {\rm L}^{s}(M)}\, < \,\varepsilon \end{equation*}

    with \varepsilon = \varepsilon(\sigma, p, N, C_{s, p}, C_p, q) sufficiently small, then

    \begin{equation} \|u(t)\|_{L^q(M)}\le \|u_{0}\|_{L^q(M)}\quad for\; all\,\, t > 0\,. \end{equation} (2.14)

    Remark 2.8. It is again possible to give an explicit estimate on the smallness parameter \varepsilon_1 above. In fact, let q_0 > 1 be fixed and \{q_m\}_{m\in\mathbb{N}} be the sequence defined by:

    q_m = p+q_{m-1}-2, \quad for\; all\,\,m\in\mathbb{N},

    so that

    \begin{equation} q_m = q_0+m(p-2). \end{equation} (2.15)

    Clearly, \{q_m\} is increasing and q_m \longrightarrow +\infty as m\to +\infty. Fix q\in[q_0, +\infty) and let \bar m be the first index such that q_{\bar m}\ge q . Define \tilde\varepsilon_1 = \tilde\varepsilon_1(\sigma, p, N, C_{s, p}, C_p, q, q_0) such that

    \begin{equation*} \label{eq71} \begin{aligned} \tilde\varepsilon_1: = \min &\left\{\left[\min\limits_{m = 0,...,\bar m}\left(\frac{p( q_m-1)^{1/p}}{p+q_m-2}\right)^pC\right]^{\frac{\sigma+p+q_m-2}{\sigma(\sigma+q_m-1)-p(p+q_m-2)}};\right.\\ &\quad\left.\left[\left(\frac{p( \sigma\frac N{p}-1)^{1/p}}{(p+\sigma\frac{N}{p}-2)}\right)^pC\right]^{\frac{\sigma+p+\sigma\frac N{p}-2}{\sigma(\sigma+\sigma\frac N{p}-1)-p(p+\sigma\frac N{p}-2)}}\right\} \end{aligned} \end{equation*}

    where C = \tilde C\, C_p^{p(\frac{p-1}{\sigma})} and \tilde C = \tilde C(C_s, p, \sigma, q) > 0 is defined in (3.37). Observe that \tilde\varepsilon_1 depends on q through the sequence \{q_m\} . More precisely, \bar m is increasing with respect to q , while the quantity \min_{m = 0, ..., \bar m}\left(\frac{p(q_m-1)^{1/p}}{p+q_m-2}\right)^pC decreases w.r.t. q_m . Furthermore, let \delta_1 > 0 be such that

    \tilde C\, \delta_1^{\frac{ps(\sigma-1)}{N(p-2)+ps}}\,+\,\frac{C\,\tilde C}{4}\delta_1^{\frac{ps(\sigma-1)}{N(p-2)+pq}} < 1\,,

    where C > 0 and \tilde C > 0 are defined in Proposition 3.3 and Lemma 4.3, respectively. Then, let q_0 = s with s as in Theorem 2.7 and define

    \varepsilon_1 = \varepsilon_1(\sigma, p, N, C_{s,p}, C_p, q, s) = \min\left\{\tilde \varepsilon_1(\sigma, p, N, C_{s,p}, C_p, q, s)\,;\delta_1\right\}\,.

    Let x_0, x \in M . We denote by r(x) = {\rm{dist}}\, (x_0, x) the Riemannian distance between x_0 and x . Moreover, we let B_R(x_0): = \{x\in M\, : {\rm{dist}}\, (x_0, x) < R\} be the geodesic ball with centre x_0 \in M and radius R > 0 . If a reference point x_0\in M is fixed, we shall simply denote by B_R the ball with centre x_0 and radius R . We also recall that \mu denotes the Riemannian measure on M .

    For any given function v , we define for any k\in {\mathbb{R}}^+

    \begin{equation} \begin{aligned} &T_k(v): = \begin{cases} k\quad &{\text{if}}\,\,\, v\ge k\,, \\ v \quad &{\text{if}}\,\,\, |v| < k\,, \\ -k\quad &{\text{if}}\,\,\, v\le -k\,;\end{cases}. \end{aligned} \end{equation} (3.1)

    For every R > 0 , k > 0, consider the problem

    \begin{equation} \begin{cases} u_t = {{\rm{div}}} \left(|\nabla u|^{p-2}\nabla u \right)+\, T_k(u^{\sigma}) \quad&{\text{in}}\,\, B_R\times (0,+\infty) \\ u = 0 &{\text{in}}\,\, \partial B_R\times (0,+\infty)\\ u = u_0 &{\text{in}}\,\, B_R\times \{0\}, \\ \end{cases} \end{equation} (3.2)

    where u_0\in L^\infty(B_R) , u_0\geq 0 . Solutions to problem (3.2) are meant in the weak sense as follows.

    Definition 3.1. Let p > 1 and \sigma > p-1 . Let u_0\in L^\infty(B_R) , u_0\geq 0 . We say that a nonnegative function u is a solution to problem (3.2) if

    u\in L^{\infty}(B_R\times(0,+\infty)), \,\,\, u\in L^2\big((0, T); W^{1,p}_0(B_R)\big) \quad\quad for\; any\, T > 0,

    and for any T > 0 , \varphi \in C_c^{\infty}(B_R\times[0, T]) such that \varphi(x, T) = 0 for every x\in B_R , u satisfies the equality:

    \begin{equation*} \begin{aligned} -\int_0^T\int_{B_R} \,u\,\varphi_t\,d\mu\,dt = &- \int_0^T\int_{B_R} |\nabla u|^{p-2} \langle\nabla u, \nabla \varphi \rangle \,d\mu\,dt\,+ \int_0^T\int_{B_R} \,T_k(u^\sigma)\,\varphi\,d\mu\,dt \\ & +\int_{B_R} \,u_0(x)\,\varphi(x,0)\,d\mu. \end{aligned} \end{equation*}

    First we consider the case \sigma > \sigma_0 where \sigma_0 has been defined in (2.1). Moreover, we assume that the Sobolev inequality (1.3) holds on M .

    Lemma 3.2. Assume (1.2) and, besides, that \sigma > p-1+\frac pN. Assume that inequality (1.3) holds. Suppose that u_0\in L^{\infty}(B_R) , u_0\ge0 . Let 1 < q < \infty and assume that

    \begin{equation} \|u_0\|_{ \rm L^{\sigma_0}(B_R)}\, < \,\bar\varepsilon \end{equation} (3.3)

    with \bar\varepsilon = \bar\varepsilon(\sigma, p, q, C_{s, p}) > 0 sufficiently small. Let u be the solution of problem (3.2) in the sense of Definition 3.1, and assume that u\in C([0, T], L^q(B_R)) for any q\in (1, +\infty) , for any T > 0 . Then

    \begin{equation} \|u(t)\|_{L^q(B_R)} \le \|u_0\|_{L^q(B_R)}\quad for \;all\,\, t > 0\,. \end{equation} (3.4)

    Note that the request u\in C([0, T], L^q(B_R)) for any q\in(1, \infty) , for any T > 0 is not restrictive, since we will construct solutions belonging to that class. This remark also applies to several other intermediate results below.

    Proof. Since u_0 is bounded and T_k(u^{\sigma}) is a bounded and Lipschitz function, by standard results, there exists a unique solution of problem (3.2) in the sense of Definition 3.1. We now multiply both sides of the differential equation in problem (3.2) by u^{q-1} ,

    \begin{align*} \label{eq0} \int_{B_R}u_t\,u^{q-1}\,dx = \int_{B_R} {{\rm{div}}} \left(|\nabla u|^{p-2}\nabla u \right)\,u^{q-1} \,dx\,+ \int_{B_R} T_k(u^{\sigma})\,u^{q-1}\,dx \,. \end{align*}

    Now, we formally integrate by parts in B_R . This can be justified by standard tools, by an approximation procedure. We get

    \begin{align} \frac{1}{q}\frac{d}{dt}\int_{B_R} u^{q}\,d\mu = -(q-1)\int_{B_R} u^{q-2}\,|\nabla u|^p \,d\mu\,+ \int_{B_R} T_k(u^{\sigma})\,u^{q-1}\,d\mu \,. \end{align} (3.5)

    Observe that, thanks to Sobolev inequality (1.3), we have

    \begin{equation} \begin{aligned} \int_{B_R} u^{q-2}\,|\nabla u|^p \,d\mu& = \left(\frac{p}{p+q-2}\right)^p \int_{B_R}\left |\nabla \left(u^{\frac{p+q-2}{p}}\right)\right|^p \,d\mu\\ &\ge \left(\frac{p}{p+q-2}\right)^pC_{s,p}^p\left( \int_{B_R} u^{\frac{p+q-2}{p}\frac{pN}{N-p}}\,d\mu \right)^{\frac{N-p}{N}}\,. \end{aligned} \end{equation} (3.6)

    Moreover, the last term in the right hand side of (3.5), by using the H{ö}lder inequality with exponents \frac{N}{N-p} and \frac{N}{p} , becomes

    \begin{equation} \begin{aligned} \int_{B_R} T_k(u^{\sigma})\,u^{q-1}\,dx&\le \int_{B_R} u^{\sigma}\,u^{q-1}\,dx = \int_{B_R} u^{\sigma-p+1}\,u^{p+q-2}\,dx \\ &\le \|u(t)\|^{\sigma-p+1}_{L^{(\sigma-p+1)\frac{N}{p}}(B_R)} \|u(t)\|^{p+q-2}_{L^{(p+q-2)\frac{N}{N-p}}(B_R)}\,. \end{aligned} \end{equation} (3.7)

    Combining (3.6) and (3.7) we get

    \begin{equation} \frac{1}{q}\frac{d}{dt} \|u(t)\|^q_{L^q(B_R)}\le -\left[(q-1)\left(\frac{p}{p+q-2}\right)^pC_{s,p}^p-\|u(t)\|^{\sigma-p+1}_{L^{\sigma_0}(B_R)}\right] \|u(t)\|^{p+q-2}_{L^{(p+q-2)\frac{N}{N-p}}(B_R)} \end{equation} (3.8)

    Take T > 0 . Observe that, due to hypotheses (3.3) and the known continuity in L^{\sigma_0} of the map t\mapsto u(t) in [0, T] , there exists t_0 > 0 such that

    \|u(t)\|_{L^{\sigma_0}(B_R)}\le 2\, \bar\varepsilon\,\,\,\,\,{\text{for any}}\,\,\,\, t\in [0,t_0]\,.

    Hence (3.8) becomes, for any t\in (0, t_0] ,

    \frac{1}{q}\frac{d}{dt} \|u(t)\|^q_{L^q_{}(B_R)}\le -\left[\left(\frac{p}{p+q-2}\right)^p(q-1)C_{s,p}^p-(2\,\bar\varepsilon)^{\sigma-p+1} \right] \|u(t)\|^{p+q-2}_{L^{(p+q-2)\frac{N}{N-p}}(B_R)}\,\le 0\,,

    where the last inequality is obtained by using (3.3). We have proved that t\mapsto \|u(t)\|_{L^q(B_R)} is decreasing in time for any t\in (0, t_0] , thus

    \begin{equation} \|u(t)\|_{L^q(B_R)}\le \|u_0\|_{L^q(B_R)}\quad {\text{for any}} \,\,\,t\in (0,t_0]\,. \end{equation} (3.9)

    In particular, inequality (3.9) follows for the choice q = \sigma_0 in view of hypothesis (3.3). Hence we have

    \|u(t)\|_{L^{\sigma_0}(B_R)}\le \|u_0\|_{L^{\sigma_0}(B_R)}\, < \,\bar\varepsilon \quad {\text{for any}} \,\,\,\,t\in (0,t_0]\,.

    Now, we can repeat the same argument in the time interval (t_0, t_1] , with t_1 = 2t_0 . This can be done due to the uniform continuity of the map t\mapsto u(t) in [0, T] . Hence, we can write that

    \|u(t)\|^{\sigma-p+1}_{L^{\sigma_0}(B_R)}\le 2\, \bar\varepsilon\,\,\,\,\,{\text{for any}}\,\,\, t\in (t_0,t_1]\,.

    Thus we get

    \begin{equation*} \|u(t)\|_{L^q(B_R)}\le \|u_0\|_{L^q(B_R)}\quad {\text{for any}} \,\,\,t\in (0,t_1]\,. \end{equation*}

    Iterating this procedure we obtain that t\mapsto\|u(t)\|_{L^q(B_R)} is decreasing in [0, T] . Since T > 0 was arbitrary, the thesis follows.

    Using a Moser type iteration procedure we prove the following result:

    Proposition 3.3. Assume (1.2) and, besides, that \sigma > p-1+\frac pN . Assume that inequality (1.3) holds. Suppose that u_0\in L^{\infty}(B_R) , u_0\ge0 . Let u be the solution of problem (3.2), so that u\in C([0, T], L^q(B_R)) for any q\in (1, +\infty) , for any T > 0 . Let 1 < q_0\le q < +\infty and assume that

    \begin{equation} \|u_0\|_{L^{\sigma_0}(B_R)}\,\le\,\tilde\varepsilon_0 \end{equation} (3.10)

    for \tilde\varepsilon_0 = \tilde\varepsilon_0(\sigma, p, N, C_{s, p}, q, q_0) sufficiently small. Then there exists C(p, q_0, C_{s, p}, \tilde\varepsilon_0, N, q) > 0 such that

    \begin{equation} \|u(t)\|_{L^q(B_R)} \le C\,t^{-\gamma_q}\|u_0\|^{\delta_q}_{L^{q_0}(B_R)}\quad for \;all\,\, t > 0\,, \end{equation} (3.11)

    where

    \begin{equation} \gamma_q = \left(\frac{1}{q_0}-\frac{1}{q}\right)\frac{N\,q_0}{p\,q_0+N(p-2)}\,,\quad \delta_q = \frac{q_0}{q}\left(\frac{q+\frac{N}{p}(p-2)}{q_0+\frac{N}{p}(p-2)}\right)\,. \end{equation} (3.12)

    Proof. Let \{q_n\} be the sequence defined in (2.8). Let \bar n be the first index such that q_{\bar n}\ge q . Observe that \bar n is well defined in view of the mentioned properties of \{q_n\} , see (2.8). We start by proving a smoothing estimate from q_0 to q_{\bar n} using a Moser iteration technique (see also [2]). Afterwards, if q_{\bar n} \equiv q then the proof is complete. Otherwise, if q_{\bar n} > q then, by interpolation, we get the thesis.

    Let t > 0 , we define

    \begin{equation} r = \frac{t}{2^{\overline n}-1} , \quad t_n = (2^n-1)r\,. \end{equation} (3.13)

    Observe that t_0 = 0, \quad t_{\bar n} = t, \quad \{t_n\} is an increasing sequence w.r.t. n . Now, for any 1\le n\le \overline n , we multiply Eq (3.2) by u^{q_{n-1}-1} and integrate in B_R\times[t_{n-1}, t_{n}] . Thus we get

    \begin{aligned} \int_{t_{n-1}}^{t_{n}}\int_{B_R} u_t\,u^{q_{n-1}-1}\,d\mu\,dt & -\int_{t_{n-1}}^{t_{n}}\int_{B_R} {{\rm{div}}}\left(|\nabla u|^{p-2}\nabla u\right)\,u^{q_{n-1}-1} \,d\mu\,dt\\ & = \int_{t_{n-1}}^{t_{n}}\int_{B_R} T_k(u^\sigma)\,u^{q_{n-1}-1}\,d\mu\,dt. \end{aligned}

    Then we integrate by parts in B_R\times[t_{n-1}, t_{n}] . Due to Sobolev inequality (1.3) and assumption (3.10), we get

    \begin{equation} \begin{aligned} &\frac{1}{q_{n-1}}\left[\|u(\cdot, t_{n})\|^{q_{n-1}}_{L^{q_{n-1}}(B_R)}-\|u(\cdot, t_{n-1})\|^{q_{n-1}}_{L^{q_{n-1}}(B_R)}\right]\\&\le -\left[\left(\frac{p}{p+q_{n-1}-2}\right)^p(q_{n-1}-1)C_{s,p}^p-2\,\tilde\varepsilon_0 \right] \int_{t_{n-1}}^{t_{n}}\|u(\tau)\|^{p+q_{n-1}-2}_{L^{(p+q_{n-1}-2)\frac{N}{N-p}}(B_R)}\,d\tau, \end{aligned} \end{equation} (3.14)

    where we have made use of inequality T_k(u^{\sigma})\, \le\, u^{\sigma}. We define q_n as in (2.8), so that (p+q_{n-1}-2)\dfrac{N}{N-p} = q_{n} . Hence, in view of hypotheses (3.10) we can apply Lemma 3.2 to the integral on the right hand side of (3.14), hence we get

    \begin{equation} \begin{aligned} &\frac{1}{q_{n-1}}\left[\|u(\cdot, t_{n})\|^{q_{n-1}}_{L^{q_{n-1}}(B_R)}-\|u(\cdot, t_{n-1})\|^{q_{n-1}}_{L^{q_{n-1}}(B_R)}\right]\\&\le -\left[\left(\frac{p}{p+q_{n-1}-2}\right)^p(q_{n-1}-1)C_{s,p}^p-2\,\tilde\varepsilon_0 \right] \|u(\cdot,t_{n})\|^{p+q_{n-1}-2}_{L^{(p+q_{n-1}-2)\frac{N}{N-p}}(B_R)}|t_{n}-t_{n-1}|. \end{aligned} \end{equation} (3.15)

    Observe that

    \begin{equation} \begin{aligned} &\left \|u(\cdot, t_{n})\right\|^{q_{n-1}}_{L^{q_{n-1}}(B_R)} \ge 0,\\ & |t_{n}-t_{n-1}| = \frac{2^{n-1}\,t}{2^{\bar n}-1}. \end{aligned} \end{equation} (3.16)

    We define

    \begin{equation} d_{n-1}: = \left[\left(\frac{p}{p+q_{n-1}-2}\right)^p(q_{n-1}-1)C_{s,p}^p-2\tilde\varepsilon_0\right]^{-1}\frac{1}{q_{n-1}}. \end{equation} (3.17)

    By plugging (3.16) and (3.17) into (3.15) we get

    \|u(\cdot, t_{n})\|^{p+q_{n-1}-2}_{L^{(p+q_{n-1}-2)\frac{N}{N-p}}(B_R)}\le \frac{(2^{\bar n}-1)d_n\,}{2^{n-1}\,t}\|u(\cdot,t_{n-1})\|^{q_{n-1}}_{L^{q_{n-1}}(B_R)}.

    The latter can be rewritten as

    \begin{equation*} \label{eq412} \|u(\cdot, t_{n})\|_{L^{(p+q_{n-1}-2)\frac{N}{N-p}}(B_R)}\le \left(\frac{(2^{\bar n}-1)d_n}{2^{n-1}}\right)^{\frac{1}{p+q_{n-1}-2}}\,t^{-\frac{1}{p+q_{n-1}-2}}\|u(\cdot,t_{n-1})\|^{\frac{q_{n-1}}{p+q_{n-1}-2}}_{L^{q_{n-1}}(B_R)}. \end{equation*}

    Due to to the definition of the sequence \{q_n\} in (2.8) we write

    \begin{equation} \|u(\cdot, t_{n})\|_{L^{q_{n}}(B_R)}\le \left(\frac{(2^{\bar n}-1)d_{n-1}}{2^{n-1}}\right)^{\frac{N}{N-p}\frac{1}{q_{n}}}\,t^{-\frac{N}{N-p}\frac{1}{q_{n}}}\left\|u(\cdot,t_{n-1})\right\|^{\frac{q_{n-1}}{q_{n}}\frac{N}{N-p}}_{L^{q_{n-1}}(B_R)}. \end{equation} (3.18)

    We define

    \begin{equation} s: = \frac{N}{N-p}. \end{equation} (3.19)

    Observe that, for any 1\le n\le \bar n , we have

    \begin{equation} \begin{aligned} \left(\frac{(2^{\bar n}-1)d_{n-1}}{2^{n-1}}\right)^{s}& = \left\{\frac{2^{\bar n}-1}{2^{n-1}}\left[\left(\frac{p}{p+q_{n-1}-2}\right)^p(q_{n-1}-1)C_{s,p}^p-2\,\varepsilon\right]^{-1}\frac{1}{q_{n-1}}\right\}^{s}\\ & = \left[\frac{2^{\bar n}-1}{2^{n-1}}\frac{1}{q_{n-1}(q_{n-1}-1)\left(\dfrac{p}{p+q_{n-1}-2}\right)^pC_{s,p}^p-2\,\varepsilon q_{n-1}}\right]^{s}, \end{aligned} \end{equation} (3.20)

    and

    \begin{equation} \frac{2^{\bar n}-1}{2^{n-1}} \le 2^{\bar n+1}\,\,\,\,\,\quad{\text{for all}}\,\,\, 1\le n\le \bar n. \end{equation} (3.21)

    Consider the function

    g(x): = \left[(x-1)\left(\frac{p}{p+x-2}\right)^pC_{s,p}^p-2\,\varepsilon\right] x\,\,\,\,\,\quad{\text{for}}\,\,\,q_0\le x \le q_{\bar n},\,\,\,x\in {\mathbb{R}}.

    Observe that, due to (2.9), g(x) > 0 for any q_0\le x \le q_{\bar n} . Moreover, g has a minimum in the interval q_0\le x \le q_{\bar n} ; call \tilde x the point at which the minimum is attained. Then we have

    \begin{equation} \frac{1}{g(x)}\le \frac{1}{g(\tilde x)} \quad\quad{\text{for any }}\,\,\,q_0\le x \le q_{\bar n}. \end{equation} (3.22)

    Thanks to (3.20)–(3.22), there exist a positive constant C , where C = C(N, C_{s, p}, \tilde\varepsilon_0, \bar n, p, q_0) such that

    \begin{equation} \left(\frac{(2^{\bar n}-1)d_{n-1}}{2^{n-1}}\right)^{s} \le C\,,\quad {\text{for all}}\,\,\, 1\le n\le\bar n. \end{equation} (3.23)

    By plugging (3.19) and (3.23) into (3.18) we get, for any 1\le n\le \bar n

    \begin{equation} \|u(\cdot, t_{n})\|_{L^{q_n}(B_R)}\le C^{\frac{1}{q_n}}t^{-\frac{s}{q_{n}}}\left\|u(\cdot,t_{n-1})\right\|^{\frac{s\,q_{n-1}}{q_{n}}}_{L^{q_{n-1}}(B_R)}. \end{equation} (3.24)

    Let us set

    U_n: = \|u(\cdot,t_n)\|_{L^{q_n}(B_R)}.

    Then (3.24) becomes

    \begin{aligned} U_n&\le C^{\frac{1}{q_n}}t^{-\frac{s}{q_{n}}}U_{n-1}^{\frac{q_{n-1}s}{q_{n}}}\\ &\le C^{\frac{1}{q_n}}t^{-\frac{s}{q_{n}}}\left [ C^{\frac{s}{q_n}}t^{-\frac{s^2}{q_{n}}} U_{k-2}^{s^2\frac{q_{n-2}}{q_n}}\right] \\ &\le ...\\ &\le C^{\frac{1}{q_n}\sum\limits_{i = 0}^{n-1}s^i}t^{-\frac{s}{q_n}\sum\limits_{i = 0}^{n-1}s^i} U_0^{s^n\frac{q_0}{q_n}}. \end{aligned}

    We define

    \begin{equation} \begin{aligned} &\alpha_n: = \frac{1}{q_n}\sum\limits_{i = 0}^{n-1}s^i,\\ &\beta_n: = \frac{s}{q_n}\sum\limits_{i = 0}^{n-1}s^i = s\,\alpha_n,\\ &\delta_n: = s^n\frac{q_0}{q_n}. \end{aligned} \end{equation} (3.25)

    By substituting n with \bar n into (3.25) we get

    \begin{equation} \begin{aligned} &\alpha_{\bar n}: = \frac{N-p}{p}\frac{A}{ q_{\bar n}},\\ &\beta_{\bar n}: = \frac{N}{p}\frac{A}{q_{\bar n}},\\ &\delta_{\bar n}: = (A+1)\frac{q_0}{q_{\bar n}}. \end{aligned} \end{equation} (3.26)

    where A: = \left(\frac{N}{N-p}\right)^{\bar n}-1 . Hence, in view of (3.13) and (3.26), (3.24) with n = \bar n yields

    \begin{equation} \|u(\cdot, t)\|_{L^{q_{\bar n}}(B_R)}\le C^{\frac{N-p}{p}\frac{A}{q_{\bar n}}}\,t^{-\frac{N}{p}\frac{A}{q_{\bar n}}}\left\|u_0\right\|^{q_{0}\frac{A+1}{q_{\bar n}}}_{L^{q_{0}}(B_R)}. \end{equation} (3.27)

    We have proved a smoothing estimate from q_0 to q_{\bar n} . Observe that if q_{\bar n} = q then the thesis is proved. Now suppose that q_{\bar n} > q . Observe that q_0\le q < q_{\bar n} and define

    B: = N(p-2)A+p\,q_0(A+1).

    From (3.27) and Lemma 3.2, we get, by interpolation,

    \begin{equation} \begin{aligned} \|u(\cdot, t)\|_{L^{ q}(B_R)}&\le \|u(\cdot, t)\|_{L^{q_0}(B_R)}^{\theta}\|u(\cdot, t)\|_{L^{q_{\bar n}}(B_R)}^{1-\theta}\\ &\le \|u_0(\cdot)\|_{L^{q_0}(B_R)}^{\theta} C\,t^{-\frac{NA}{B}(1-\theta)}\left\|u_0\right\|^{pq_{0}\frac{A+1}{B}(1-\theta)}_{L^{q_{0}}(B_R)}\\ & = C\,t^{-\frac{N\,A}{B}(1-\theta)}\left\|u_0\right\|^{pq_{0}\frac{A+1}{B}(1-\theta)+\theta}_{L^{q_{0}}(B_R)}, \end{aligned} \end{equation} (3.28)

    where

    \begin{equation} \theta = \frac{q_0}{ q}\left(\frac{q_{\bar n} - q}{q_{\bar n} - q_0}\right). \end{equation} (3.29)

    Observe that

    \begin{aligned} &{\text{(i)}}\quad\frac{N A}{B}(1-\theta) = \frac{N}{p}\left(\frac{q-q_0}{q}\right)\frac{1}{q_0+\frac{N}{p}(p-2)};\\ &{\text{(ii)}}\quad p\,q_0\frac{A+1}{B}(1-\theta)+\theta = \frac{q_0}{q}\frac{q+\frac{N}{p}(p-2)}{q_0+\frac{N}{p}(p-2)}. \end{aligned}

    Combining (3.28), (3.12) and (3.29) we get the claim, noticing that q was arbitrarily in [q_0, +\infty) .

    Remark 3.4 One can not let q\to+\infty is the above bound. In fact, one can show that \varepsilon \longrightarrow 0 as q\to\infty . So in such limit the hypothesis on the norm of the initial datum (2.9) is satisfied only when u_0\equiv 0 .

    We now consider the case \sigma > p-1 and that the Sobolev and Poincaré inequalities (1.3), (1.4) hold on M .

    Lemma 3.5. Assume (1.2) and, besides, that p > 2 . Assume that inequalities (1.3) and (1.4) hold. Suppose that u_0\in L^{\infty}(B_R) , u_0\ge0 . Let 1 < q < \infty and assume that

    \begin{equation} \|u_0\|_{L^{\sigma\frac N{p}}(B_R)} < \bar\varepsilon_1 \end{equation} (3.30)

    for a suitable \tilde\varepsilon_1 = \tilde \varepsilon_1(\sigma, p, N, C_p, C_{s, p}, q) sufficiently small. Let u be the solution of problem (3.2) in the sense of Definition 3.1, such that in addition u\in C([0, T); L^q(B_R)) . Then

    \begin{equation} \|u(t)\|_{L^q(B_R)} \le \|u_0\|_{L^q(B_R)}\quad for\; all\,\, t > 0\,. \end{equation} (3.31)

    Proof. Since u_0 is bounded and T_k(u^{\sigma}) is a bounded and Lipschitz function, by standard results, there exists a unique solution of problem (3.2) in the sense of Definition 3.1. We now multiply both sides of the differential equation in problem (3.2) by u^{q-1} , therefore

    \begin{align} \int_{B_R} u_t\,u^{q-1}\,d\mu = \int_{B_R} {{\rm{div}}}(|\nabla u|^{p-2}\,\nabla u)u^{q-1}\,d\mu\,+ \int_{B_R} T_k(u^{\sigma})\,u^{q-1}\,d\mu \,. \end{align}

    We integrate by parts. This can again be justified by a standard approximation procedure. By using the fact that T(u^\sigma)\le u^\sigma , we can write

    \begin{equation} \begin{aligned} \frac{1}{q}\frac{d}{dt}\int_{B_R} u^{q}\,d\mu \le-(q-1)\left(\frac{p}{p+q-2}\right)^p\int_{B_R} \left|\nabla\left( u^{\frac{p+q-2}{p}}\right)\right|^p \,d\mu\,+ \int_{B_R} u^{\sigma+q-1}\,d\mu. \end{aligned} \end{equation} (3.32)

    Now we take c_1 > 0 , c_2 > 0 such that c_1+c_2 = 1 so that

    \begin{align} \int_{B_R} \left|\nabla\left( u^{\frac{p+q-2}{p}}\right)\right|^p \,d\mu = c_1\, \left\|\nabla\left( u^{\frac{p+q-2}{p}}\right)\right\|_{L^p(B_R)}^p \, + c_2\, \left\|\nabla\left( u^{\frac{p+q-2}{p}}\right)\right\|_{L^p(B_R)}^p. \end{align} (3.33)

    Take \alpha\in (0, 1) . Thanks to (1.4), (3.33) we get

    \begin{equation} \begin{aligned} \int_{B_R} \left|\nabla\left( u^{\frac{p+q-2}{p}}\;\right)\right|^2 \,d\mu& \ge c_1\,C_p^p \left\| u\right\|^{p+q-2}_{L^{p+q-2}\;\;(B_R)}\, + c_2\, \left\|\nabla\left( u^{\frac{p+q-2}{p}}\right)\right\|_{L^p(B_R)}^p\\ &\ge c_1C_p^p \left\| u\right\|^{p+q-2}_{L^{p+q-2}\;\;(B_R)}\, +c_2 \left\|\nabla\left( u^{\frac{p+q-2}{p}}\right)\right\|_{L^p(B_R)}^{p+p\alpha-p\alpha}\\ &\ge c_1C_p^p \left\| u\right\|^{p+q-2}_{L^{p+q-2}\;\;(B_R)}\, + c_2C_p^{p\alpha} \left\| u\right\|^{\alpha(p+q-2)}_{L^{p+q-2}\;\;(B_R)} \left\|\nabla\left( u^{\frac{p+q-2}{p}}\right)\right\|_{L^p(B_R)}^{p-p\alpha} \end{aligned} \end{equation} (3.34)

    Moreover, using the interpolation inequality, Hölder inequality and (1.3), we have

    \begin{equation} \begin{aligned} \int_{B_R} u^{\sigma+q-1}\,d\mu,& = \|u\|_{L^{\sigma+q-1}}^{\sigma+q-1}\\ &\le \|u\|_{L^{p+q-2}\;\;(B_R)}^{\theta(\sigma+q-1)}\,\|u\|_{L^{\sigma+p+q-2}(B_R)}^{(1-\theta)(\sigma+q-1)}\\ &\le \|u\|_{L^{p+q-2}\;\;(B_R)}^{\theta(\sigma+q-1)}\left[\|u\|_{L^{\sigma\frac{N}{p}}(B_R)}^{\sigma}\|u\|_{L^{(p+q-2)\frac{N}{N-p}}\;(B_R)}^{p+q-2}\right]^{\frac{(1-\theta)(\sigma+q-1)}{\sigma+p+q-2}}\\ &\le \|u\|_{L^{p+q-2}\;\;(B_R)}^{\theta(\sigma+q-1)}\|u\|_{L^{\sigma\frac{N}{p}}(B_R)}^{(1-\theta)\frac{\sigma(\sigma+q-1)}{\sigma+p+q-2}} \;\;\left(\frac{1}{C_{s,p}}\left\|\nabla \left(u^{\frac{p+q-2}{p}}\;\right)\right\|_{L^p(B_R)}\right)^{p(1-\theta)\frac{\sigma+q-1}{\sigma+p+q-2}} \end{aligned} \end{equation} (3.35)

    where \theta: = \frac{(p-1)(p+q-2)}{\sigma(\sigma+q-1)} . By plugging (3.34) and (3.35) into (3.32) we obtain

    \begin{equation} \begin{aligned} \frac{1}{q}\frac{d}{dt}\|u(t)\|_{L^q(B_R)}^{q} & \le-(q-1)\left(\frac{p}{p+q-2}\right)^p c_1\,C_p^p \left\| u\right\|^{p+q-2}_{L^{p+q-2}\;\;(B_R)}\, \\ & - (q-1)\left(\frac{p}{p+q-2}\right)^p c_2\,C_p^{p\alpha} \left\| u\right\|^{\alpha(p+q-2)}_{L^{p+q-2}\;\;(B_R)} \left\|\nabla\left( u^{\frac{p+q-2}{p}}\right)\right\|_{L^p(B_R)}^{p-p\alpha} \\ & +\tilde{C}\|u\|_{L^{p+q-2}\;\;(B_R)}^{\theta(\sigma+q-1)}\,\|u\|_{L^{\sigma\frac{N}{p}}(B_R)}^{(1-\theta)\frac{\sigma(\sigma+q-1)}{\sigma+p+q-2}} \|\nabla \left(u^{\frac{p+q-2}{p}}\right)\|_{L^p(B_R)}^{p(1-\theta)\frac{\sigma+q-1}{\sigma+p+q-2}}, \end{aligned} \end{equation} (3.36)

    where

    \begin{equation} \tilde{C} = \left(\frac{1}{C_{s,p}}\right)^{p(1-\theta)\frac{\sigma+q-1}{\sigma+p+q-2}}. \end{equation} (3.37)

    Let us now fix \alpha\in (0, 1) such that

    p-p\alpha = p(1-\theta)\frac{\sigma+q-1}{\sigma+p+q-2}.

    Hence, we have

    \begin{equation} \alpha = \frac{p-1}{\sigma}. \end{equation} (3.38)

    By substituting (3.38) into (3.36) we obtain

    \begin{equation} \begin{aligned} \frac{1}{q}\frac{d}{dt}\|u(t)\|_{L^q(B_R)}^{q} &\le -(q-1)\left(\frac{p}{p+q-2}\right)^p c_1\,C_p^p \left\| u\right\|^{p+q-2}_{L^{p+q-2}\;\;(B_R)}\\ & - \frac{1}{\tilde C}\left\{ (q-1)\left(\frac{p}{p+q-2}\right)^pC - \|u\|_{L^{\sigma\frac{N}{p}}(B_R)}^{\frac{\sigma(\sigma+q-1)-(p-1)(p+q-2)}{\sigma+p+q-2}}\;\;\right\} \\ &\times\left\| u\right\|^{\alpha(p+q-2)}_{L^{p+q-2}\;\;(B_R)} \left\|\nabla\left( u^{\frac{p+q-2}{p}}\right)\right\|_{L^p(B_R)}^{p-p\alpha}, \end{aligned} \end{equation} (3.39)

    where C has been defined in Remark 2.8. Observe that, due to hypotheses (3.30) and by the continuity of the solution u(t) , there exists t_0 > 0 such that

    \left\| u(t)\right\|_{L^{\sigma\frac N{p}}(B_R)}\le 2\, \tilde\varepsilon_1\,\,\,\,\,{\text{for any}}\,\,\,\, t\in (0,t_0]\,.

    Hence, (3.39) becomes, for any t\in (0, t_0]

    \begin{equation*} \begin{aligned} \frac{1}{q}\frac{d}{dt}\|u(t)\|_{L^q(B_R)}^{q} &\le -(q-1)\left(\frac{p}{p+q-2}\right)^p c_1C_p^p \left\| u\right\|^{p+q-2}_{L^{p+q-2}\;\;(B_R)}\, \\ & - \frac{1}{\tilde C}\left\{ (q-1)\left(\frac{p}{p+q-2}\right)^pC -2\tilde \varepsilon_1^{\frac{\sigma(\sigma+q-1)-(p-1)(p+q-2)}{\sigma+p+q-2}}\right\} \left\| u\right\|^{\alpha(p+q-2)}_{L^{p+q-2}\;\;(B_R)} \left\|\nabla\left( u^{\frac{p+q-2}{p}}\right)\right\|_{L^p(B_R)}^{p-p\alpha}\\ &\\ &\le 0\,, \end{aligned} \end{equation*}

    provided \tilde\varepsilon_1 is small enough. Hence we have proved that \|u(t)\|_{L^q(B_R)} is decreasing in time for any t\in (0, t_0] , thus

    \begin{equation} \|u(t)\|_{L^q(B_R)}\le \|u_0\|_{L^q(B_R)}\quad {\text{for any}} \,\,\,t\in (0,t_0]\,. \end{equation} (3.40)

    In particular, inequality (3.40) holds q = \sigma\frac N{p} . Hence we have

    \|u(t)\|_{L^{\sigma\frac N{p}}(B_R)}\le \|u_0\|_{L^{\sigma\frac N{p}}(B_R)}\, < \,\tilde\varepsilon_1\quad {\text{for any}} \,\,\,\,t\in (0,t_0]\,.

    Now, we can repeat the same argument in the time interval (t_0, t_1] with t_1 = 2t_0 . This can be done due to the uniform continuity of the map t\mapsto u(t) in [0, T] . Hence, we can write that

    \left\| u(t)\right\|_{L^{\sigma\frac N{p}}(B_R)}\le 2\, \tilde\varepsilon_1\,\,\,\,\,{\text{for any}}\,\,\, t\in (t_0,t_1]\,.

    Thus we get

    \begin{equation*} \|u(t)\|_{L^{q}(B_R)}\le \|u_0\|_{L^q(B_R)}\quad {\text{for any}} \,\,\,t\in (0,t_1]\,. \end{equation*}

    Iterating this procedure we obtain the thesis.

    Using a Moser type iteration procedure we prove the following result:

    Proposition 3.6. Assume (1.2) and, besides, that p > 2 . Let M be such that (1.3) and (1.4) hold. Suppose that u_0\in L^{\infty}(B_R) , u_0\ge0 . Let u be the solution of problem (3.2) in the sense of Definition 3.1 such that in addition u\in C([0, T], L^q(B_R)) for any q\in(1, +\infty) , for any T > 0 . Let 1 < q_0\le q < +\infty and assume that

    \begin{equation} \|u_0\|_{L^{\sigma\frac N{p}}}(B_R) < \tilde{\varepsilon}_1 \end{equation} (3.41)

    for \tilde{\varepsilon}_1 = \tilde{\varepsilon}_1(\sigma, p, N, C_{s, p}, C_p, q, q_0) sufficiently small. Then there exists C(p, q_0, C_{s, p}, \tilde\varepsilon_1, N, q) > 0 such that

    \begin{equation} \|u(t)\|_{L^q(B_R)} \le C\,t^{-\gamma_q}\|u_0\|^{\delta_q}_{L^{q_0}(B_R)}\quad for \;all\,\, t > 0\,, \end{equation} (3.42)

    where

    \begin{equation} \gamma_q = \frac{q_0}{p-2}\left(\frac{1}{q_0}-\frac{1}{q}\right)\,,\quad \delta_q = \frac{q_0}{q}\,. \end{equation} (3.43)

    Proof. Arguing as in the proof of Proposition 3.3, let \{q_m\} be the sequence defined in (2.15). Let \overline m be the first index such that q_{\overline m}\ge q . Observe that \bar m is well defined in view of the mentioned properties of \{q_m\} , see (2.15). We start by proving a smoothing estimate from q_0 to q_{\overline m} using again a Moser iteration technique. Afterwards, if q_{\overline m} \equiv q then the proof is complete. Otherwise, if q_{\overline m} > q then, by interpolation, we get the thesis.

    Let t > 0 , we define

    \begin{equation} r = \frac{t}{2^{\overline m}-1} , \quad t_m = (2^m-1)r\,. \end{equation} (3.44)

    Observe that

    t_0 = 0, \quad t_{\overline m} = t,\quad \{t_m\}\,{\text{ is an increasing sequence w.r.t.}}\,\,m.

    Now, for any 1\le m\le \overline m , we multiply Eq (3.2) by u^{q_{m-1}-1} and integrate in B_R\times[t_{m-1}, t_{m}] . Thus we get

    \begin{aligned} \int_{t_{m-1}}^{t_{m}}\int_{B_R}u_t\,u^{q_{m-1}-1}\,d\mu\,d\tau &-\int_{t_{m-1}}^{t_{m}}\int_{B_R} {{\rm{div}}}\left(|\nabla u^{p-2}|\nabla u\right)\,u^{q_{m-1}-1} \,d\mu\,d\tau\\ &\,\,\, = \int_{t_{m-1}}^{t_{m}}\int_{B_R} T_k(u^\sigma)\,u^{q_{m-1}-1}\,d\mu\,d\tau. \end{aligned}

    Then we integrate by parts in B_R\times[t_{m-1}, t_{m}] , hence we get

    \begin{equation*} \label{eq718} \begin{aligned} \frac{1}{q_{m-1}}&\left[\|u(\cdot, t_{m})\|^{q_{m-1}}_{L^{q_{m-1}}(B_R)}-\|u(\cdot, t_{m-1})\|^{q_{m-1}}_{L^{q_{m-1}}(B_R)}\right]\\&\le - (q_{m-1}-1)\left(\frac{p}{p+q_{m-1}-2}\right)^p\int_{t_{m-1}}^{t_{m}} \int_{B_R} \left|\nabla\left( u^{\frac{p+q_{m-1}-2}{p}}\right)\right|^p \,d\mu\,d\tau\\ &\quad\quad+\int_{t_{m-1}}^{t_{m}}\int_{B_R} u^\sigma\,u^{q_{m-1}-1}\,d\mu\,d\tau. \end{aligned} \end{equation*}

    where we have made use of inequality

    T_k(u^\sigma)\,\le\,u^\sigma.

    Now, by arguing as in the proof of Lemma 3.5, by using (3.33) and (3.34) with q = q_{m-1} , we get

    \begin{equation*} \label{eq719} \begin{aligned} &\int_{B_R} \left|\nabla\left( u^{\frac{p+q_{m-1}-2}{p}}\right)\right|^pd\mu\\ &\quad\quad\quad \ge c_1C_p^p \left\| u\right\|^{p+q_{m-1}-2}_{L^{p+q_{m-1}-2}(B_R)} + c_2C_p^{p\alpha} \left\| u\right\|^{\alpha(p+q_{m-1}-2)}_{L^{p+q_{m-1}-2}(B_R)} \left\|\nabla\left( u^{\frac{p+q_{m-1}-2}{p}}\right)\right\|_{L^p(B_R)}^{p-p\alpha} \end{aligned} \end{equation*}

    where \alpha\in(0, 1) and c_1 > 0 , c_2 > 0 with c_1+c_2 = 1 . Similarly, from (3.35) with q = q_{m-1} we can write

    \begin{equation*} \label{eq720} \begin{aligned} \int_{B_R}u^\sigma u^{q_{m-1}-1}d\mu& = \|u\|_{L^{p+q_{m-1}-1}(B_R)}^{\sigma+q_{m-1}-1}\\ &\le \|u\|_{L^{p+q_{m-1}-2}(B_R)}^{\theta(\sigma+q_{m-1}-1)}\,\|u\|_{L^{\sigma\frac{N}{p}}(B_R)}^{(1-\theta)\frac{\sigma(\sigma+q_{m-1}-1)}{\sigma+p+q_{m-1}-2}} \\ &\quad \times\left(\frac{1}{C_{s,p}}\left\|\nabla(u^{\frac{p+q_{m-1}-2}{p}})\right\|_{L^p(B_R)}\right)^{p(1-\theta)\frac{\sigma+q_{m-1}-1}{\sigma+p+q_{m-1}-2}} \end{aligned} \end{equation*}

    where \theta: = \frac{(p-1)(p+q_{m-1}-2)}{\sigma(\sigma+q_{m-1}-1)} . Now, due to assumption (3.30), the continuity of u , by choosing \tilde C and \alpha as in (3.37) and (3.38) respectively, we can argue as in the proof of Lemma 3.5 (see (3.39)), hence we obtain

    \begin{equation} \begin{aligned} \frac{1}{q_{m-1}}&\left[\|u(\cdot, t_{m})\|^{q_{m-1}}_{L^{q_{m-1}}(B_R)}-\|u(\cdot, t_{m-1})\|^{q_{m-1}}_{L^{q_{m-1}}(B_R)}\right]\\ &\le-(q_{m-1}-1)\left(\frac{p}{p+q_{m-1}-2}\right)^pc_1C_p^p \int_{t_{m-1}}^{t_{m}} \left\| u(\cdot, \tau)\right\|^{p+q_{m-1}-2}_{L^{p+q_{m-1}-2}(B_R)} d\tau \\ & - \frac{1}{\tilde C}\left\{(q_{m-1}-1)\left(\frac{p}{p+q_{m-1}-2}\right)^pC\, - 2\tilde{\varepsilon_1}^{\frac{\sigma(\sigma+q_{m-1}-1)-(p-1)(p+q_{m-1}-2)}{\sigma+p+q_{m-1}-2}}\right\} \\ &\times \int_{t_{m-1}}^{t_m}\left\| u(\cdot,\tau)\right\|^{\alpha(p+q_{m-1}-2)}_{L^{p+q_{m-1}-2}(B_R)} \left\|\nabla\left( u^{\frac{p+q_{m-1}-2}{p}}\right)(\cdot,\tau)\right\|_{L^p(B_R)}^{p-p\alpha}\,d\tau, \end{aligned} \end{equation} (3.45)

    where C has been defined in Remark 2.8. Finally, provided \tilde\varepsilon_1 is small enough, (3.45) can be rewritten as

    \begin{equation*} \begin{aligned} \frac{1}{q_{m-1}}&\left[\|u(\cdot, t_{m})\|^{q_{m-1}}_{L^{q_{m-1}}(B_R)}-\|u(\cdot, t_{m-1})\|^{q_{m-1}}_{L^{q_{m-1}}(B_R)}\right]\\ &\quad\quad\quad\le-(q_{m-1}-1)\left(\frac{p}{p+q_{m-1}-2}\right)^pc_1C_p^p \int_{t_{m-1}}^{t_{m}} \left\| u(\cdot, \tau)\right\|^{p+q_{m-1}-2}_{L^{p+q_{m-1}-2}(B_R)} d\tau. \end{aligned} \end{equation*}

    We define q_m as in (2.15), so that q_m = p+q_{m-1}-2 . Then, in view of hypothesis (3.41), we can apply Lemma 3.5 to the integral in the right-hand side of the latter, hence we get

    \begin{equation} \begin{aligned} \frac{1}{q_{m-1}}&\left[\|u(\cdot, t_{m})\|^{q_{m-1}}_{L^{q_{m-1}}(B_R)}-\|u(\cdot, t_{m-1})\|^{q_{m-1}}_{L^{q_{m-1}}(B_R)}\right]\\ &\quad\quad\quad\le-(q_{m-1}-1)\left(\frac{p}{p+q_{m-1}-2}\right)^pc_1C_p^p\left\| u(\cdot,t_m)\right\|^{q_m}_{L^{q_m}(B_R)} |t_m-t_{m-1}|. \end{aligned} \end{equation} (3.46)

    Observe that

    \begin{equation} \begin{aligned} &\|u(\cdot, t_{m})\|^{q_{m-1}}_{L^{q_{m-1}}(B_R)}\,\ge\,0,\\ &|t_m-t_{m-1}| = \frac{2^{m-1}t}{2^{\overline m}-1}. \end{aligned} \end{equation} (3.47)

    We define

    \begin{equation} d_{m-1}: = \left(\frac{p}{p+q_{m-1}-2}\right)^{-p}\frac 1{c_1\,C_p^p}\frac{1}{q_{m-1}(q_{m-1}-1)}. \end{equation} (3.48)

    By plugging (3.47) and (3.48) into (3.46), we get

    \left\| u(\cdot,t_m)\right\|^{q_m}_{L^{q_m}_{\rho}(B_R)}\,\le \,\frac{2^{\bar m}-1}{2^{m-1}t}\,d_{m-1}\|u(\cdot,t_{m-1})\|^{q_{m-1}}_{L^{q_{m-1}}_{\rho}(B_R)}.

    The latter can be rewritten as

    \begin{equation} \left\| u(\cdot,t_m)\right\|_{L^{q_m}(B_R)}\,\le \,\left(\frac{2^{\bar m}-1}{2^{m-1}}\,d_{m-1}\right)^{\frac 1{q_m}}t^{-\frac1{q_m}}\|u(\cdot,t_{m-1})\|^{\frac{q_{m-1}}{q_m}}_{L^{q_{m-1}}(B_R)} \end{equation} (3.49)

    Observe that, for any 1\le m\le \bar m , we have

    \begin{equation} \begin{aligned} \frac{2^{\bar m}-1}{2^{m-1}}\,d_{m-1}& = \frac{2^{\bar m}-1}{2^{m-1}}\left(\frac{p}{p+q_{m-1}-2}\right)^{-p}\frac 1{c_1\,C_p^p}\frac{1}{q_{m-1}(q_{m-1}-1)}\\ &\le 2^{\bar m+1}\frac{1}{c_1\,C_p^p}\left(\frac{p+q_{m-1}-2}{p}\right)^{p}\frac{1}{q_{m-1}(q_{m-1}-1)}. \end{aligned} \end{equation} (3.50)

    Consider the function

    h(x): = \frac {(p+x-2)^p}{x(x-1)}, \quad {\text{for}}\,\,\,q_0\le x\le q_{\overline m},\quad x\in {\mathbb{R}}.

    Observe that h(x)\ge0 for any q_0\le x\le q_{\overline m} . Moreover, h has a maximum in the interval q_0\le x\le q_{\overline m} , call \tilde{x} the point at which it is attained. Hence

    \begin{equation} h(x)\le h(\tilde x)\quad {\text{for any}}\,\,\,q_0\le x\le q_{\overline m},\quad x\in {\mathbb{R}}. \end{equation} (3.51)

    Due to (3.50) and (3.51), we can say that there exists a positive constant C , where C = C(C_p, \bar m, p, q_0) , such that

    \begin{equation} \frac{2^{\overline m}-1}{2^{m-1}}\,d_{m-1}\le C\quad {\text{for all}}\,\,1\le m\le \overline m. \end{equation} (3.52)

    By using (3.52) and (3.49), we get, for any 1\le m\le \overline m

    \begin{equation} \left\| u(\cdot,t_m)\right\|_{L^{q_m}(B_R)}\,\le\, C^{\frac1{q_m}}t^{-\frac1{q_m}}\|u(\cdot,t_{m-1})\|^{\frac{q_{m-1}}{q_m}}_{L^{q_{m-1}}(B_R)}. \end{equation} (3.53)

    Let us set

    U_m: = \left\| u(\cdot,t_m)\right\|_{L^{q_m}(B_R)}

    Then (3.53) becomes

    \begin{aligned} U_m&\le C^{\frac1{q_m}}t^{-\frac1{q_m}}U_{n-1}^{\frac{q_{m-1}}{q_m}}\\ &\le C^{\frac1{q_m}}t^{-\frac1{q_m}}\left[C^{\frac1{q_{m-1}}}t^{-\frac1{q_{m-1}}}U_{m-2}^{\frac{q_{m-2}}{q_{m-1}}}\right]\\ &\le ...\\ &\le C^{\frac m{q_m}}t^{-\frac m{q_m}}U_0^{\frac{q_0}{q_m}}. \end{aligned}

    We define

    \begin{equation} \alpha_m: = \frac m{q_m},\quad \delta_m: = \frac{q_0}{q_m}. \end{equation} (3.54)

    Substituting m with \bar m into (3.54) and in view of (3.44), (3.53) with m = \overline m , we have

    \begin{equation*} \label{eq732} \left\| u(\cdot,t)\right\|_{L^{q_{\overline m}}(B_R)}\,\le\, C^{\alpha_{\overline m}}t^{-\alpha_{\overline m}}\left\| u_0\right\|_{L^{q_{0}}(B_R)}^{\delta_{\overline m}}. \end{equation*}

    Observe that if q_{\overline m} = q then the thesis is proved and one has

    \alpha_{\overline m} = \frac1{p-2}\left(1-\frac{q_0}{q}\right),\quad \delta_{\overline m} = \frac{q_0}{q}.

    Now suppose that q < q_{\overline m} , then in particular q_0\le q\le q_{\overline m} . By interpolation and Lemma 3.5 we get

    \begin{equation} \begin{aligned} \left\| u(\cdot,t)\right\|_{L^{q}(B_R)}&\le \left\| u(\cdot,t)\right\|_{L^{q_{0}}(B_R)}^{\theta}\left\| u(\cdot,t)\right\|_{L^{q_{\overline m}}(B_R)}^{1-\theta}\\ & \left\| u(\cdot,t)\right\|_{L^{q_{0}}(B_R)}^{\theta}\, C^{\alpha_{\overline m}(1-\theta)}t^{-\alpha_{\overline m}(1-\theta)}\left\| u_0\right\|_{L^{q_{0}}(B_R)}^{\delta_{\overline m}(1-\theta)}\\ &\le C^{\alpha_{\overline m}(1-\theta)}t^{-\alpha_{\overline m}(1-\theta)}\left\| u_0\right\|_{L^{q_{0}}(B_R)}^{\delta_{\overline m}(1-\theta)+\theta}, \end{aligned} \end{equation} (3.55)

    where

    \begin{equation} \theta = \frac{q_0}{q}\left(\frac{q_{\overline m}-q}{q_{\overline m}-q_0}\right). \end{equation} (3.56)

    Combining (3.43), (3.55) and (3.56), we get the claim by noticing that q was arbitrary fixed in [q_0, +\infty) .

    In what follows, we will deal with solutions u_R to problem (3.2) for arbitrary fixed R > 0 . For notational convenience, we will simply write u instead of u_R since no confusion will occur in the present section. We define

    \begin{equation} G_k(v): = v-T_k(v). \end{equation} (4.1)

    where T_k(v) has been defined in (3.1). Let a_1 > 0 , a_2 > 0 and t > \tau_1 > \tau_2 > 0 . We consider, for any i\in\mathbb N\cup\{0\} , the sequences

    \begin{equation} \begin{aligned} &k_i: = a_2+(a_1-a_2)2^{-i}\,;\\ &\theta_i: = \tau_2+(\tau_1-\tau_2)2^{-i}\,; \end{aligned} \end{equation} (4.2)

    and the cylinders

    \begin{equation} U_i: = B_R\times(\theta_i,t). \end{equation} (4.3)

    Observe that the sequence \{\theta_i\}_{i\in \mathbb{N}} is monotone decreasing w.r.t. i . Furthermore, we define, for any i\in\mathbb{N} , the cut-off functions \xi_i(\tau) such that

    \begin{equation} \xi_i(\tau): = \begin{cases} 1\quad &\theta_{i-1} < \tau < t\\ 0\quad &0 < \tau < \theta_i \end{cases}\quad\quad{\text{and}}\quad\quad|(\xi_i)_{\tau}|\,\le\, \frac{2^i}{\tau_1-\tau_2}\,. \end{equation} (4.4)

    Finally, we define

    \begin{equation} S(t): = \sup\limits_{0 < \tau < t}\left(\tau\|u(\tau)\|_{L^{\infty}(B_R)}^{\sigma-1}\right). \end{equation} (4.5)

    We can now state the following

    Lemma 4.1. Let i\in\mathbb{N} , k_i , \theta_i , U_i be defined in (4.2), (4.3) and R > 0 . Let u be a solution to problem (3.2). Then, for any q > 1 , we have thatX

    \begin{align*} \label{eq35} \sup\limits_{\tau_1 < \tau < t}\int_{B_R}[G_{k_0}(u)]^q\,d\mu + \iint_{U_{i-1}}\left|\nabla[G_{k_i}(u)]^{\frac{p+q-2}{p}}\right|^p\,d\mu d\tau\le 2^i\gamma\,C_1\iint_{U_i}[G_{k_{i+1}}(u)]^q\,d\mu d\tau. \end{align*}

    where \gamma = \gamma(p, q) and

    \begin{equation} C_1: = \frac{1}{\tau_1-\tau_2}+\frac{S(t)}{\tau_1}\frac{2a_1}{a_1-a_2}. \end{equation} (4.6)

    Proof. For any i\in\mathbb{N} , we multiply both sides of the differential equation in problem (3.2) by [G_{k_i}(u)]^{q-1}\xi_i , q > 1 , and we integrate on the cylinder U_i , yielding:

    \begin{equation} \begin{aligned} \iint_{U_i} &u_{\tau}\,[G_{k_i}(u)]^{q-1}\xi_i\,d\mu d\tau \\ & = \iint_{U_i} {{\rm{div}}}(|\nabla u|^{p-2}\,\nabla u)[G_{k_i}(u)]^{q-1}\xi_i\,d\mu d\tau\,+ \iint_{U_i} T_k(u^{\sigma})\,[G_{k_i}(u)]^{q-1}\xi_i\,d\mu d\tau\,. \end{aligned} \end{equation} (4.7)

    We integrate by parts. Thus we write, due to (4.4),

    \begin{equation} \begin{aligned} \iint_{U_i} u_{\tau}\,[G_{k_i}(u)]^{q-1}\xi_i\,d\mu d\tau & = \frac 1q \iint_{U_i} \frac{d}{d\tau}[(G_{k_i}(u))^{q}]\xi_i\,d\mu d\tau\\ & = -\frac 1q \iint_{U_i}[G_{k_i}(u)]^{q}(\xi_i)_{\tau}\,d\mu d\tau+\frac 1q\int_{B_R}[G_{k_i}(u(x,t))]^{q}\,d\mu \end{aligned} \end{equation} (4.8)

    Moreover,

    \begin{equation} \begin{aligned} -\iint_{U_i} {{\rm{div}}}(|\nabla u|^{p-2}\,\nabla u)[G_{k_i}(u)]^{q-1}&\xi_i\,d\mu d\tau = \iint_{U_i}|\nabla u|^{p-2}\,\nabla u\cdot \nabla[G_{k_i}(u)]^{q-1}\xi_i\,d\mu d\tau\\ &\ge(q-1) \iint_{U_i} [G_{k_i}(u)]^{q-2} |\nabla [G_{k_i}(u)] |^{p}\, \xi_i\,d\mu d\tau. \end{aligned} \end{equation} (4.9)

    Now, combining (4.7), (4.8) and (4.9), using the fact that T(u^\sigma)\le u^\sigma and (4.4), we can write

    \begin{equation} \begin{aligned} \frac 1q\int_{B_R}[G_{k_i}(u(x,t))]^{q}\,d\mu &+(q-1) \iint_{U_i} [G_{k_i}(u)]^{q-2} |\nabla [G_{k_i}(u)] |^{p}\, \xi_i\,d\mu d\tau\\ &\le \frac 1q \iint_{U_i}[G_{k_i}(u)]^{q}(\xi_i)_{\tau}\,d\mu d\tau+\iint_{U_i} u^{\sigma}\,[G_{k_i}(u)]^{q-1}\xi_i\,d\mu d\tau\,\\ &\le \frac{2^i}{\tau_1-\tau_2}\iint_{U_i}[G_{k_i}(u)]^{q}\,d\mu d\tau+\iint_{U_i} u^{\sigma}\,[G_{k_i}(u)]^{q-1}\xi_i\,d\mu d\tau. \end{aligned} \end{equation} (4.10)

    Let us define

    \tilde\gamma: = \left[\min\left\{\frac 1q,\,q-1\right\}\right]^{-1},

    thus (4.10) reads

    \begin{equation} \begin{aligned} \int_{B_R}[G_{k_i}(u(x,t))]^{q}\,d\mu &+ \iint_{U_{i}} [G_{k_i}(u)]^{q-2} |\nabla [G_{k_i}(u)] |^{p}\xi_i\,d\mu d\tau\\ &\le \tilde\gamma \frac{2^i}{\tau_1-\tau_2}\iint_{U_i}[G_{k_i}(u)]^{q}\,d\mu d\tau+\tilde\gamma \iint_{U_i} u^{\sigma}\,[G_{k_i}(u)]^{q-1}\xi_id\mu d\tau. \end{aligned} \end{equation} (4.11)

    Observe that the sequence \{k_i\}_{i\in\mathbb{N}} is monotone decreasing, hence

    G_{k_0}(u)\le G_{k_i}(u)\le G_{k_{i+1}}(u)\le u\quad\quad{\text{for all}}\,\,\,i\in\mathbb{N}.

    Thus (4.11) can be rewritten as

    \begin{equation} \begin{aligned} \int_{B_R}[G_{k_0}(u(x,t))]^{q}\,d\mu &+ \iint_{U_{i-1}} [G_{k_i}(u)]^{q-2} |\nabla [G_{k_i}(u)] |^{p}\,d\mu d\tau\\ &\le \frac{2^i\,\tilde\gamma }{\tau_1-\tau_2}\iint_{U_i}[G_{k_{i+1}}(u)]^{q}\,d\mu d\tau+\tilde\gamma \iint_{U_i} u^{\sigma}\,[G_{k_{i+1}}(u)]^{q-1}d\mu d\tau. \end{aligned} \end{equation} (4.12)

    Let us now define

    I: = \tilde\gamma \iint_{U_i} u^{\sigma-1}\,u\,[G_{k_{i+1}}(u)]^{q-1}d\mu d\tau

    Observe that, for any i\in\mathbb{N} ,

    \frac{u}{k_i}\chi_i\,\le\, \frac{u-k_{i+1}}{k_i-k_{i+1}}\chi_i

    where \chi_i is the characteristic function of D_i: = \{(x, t)\in U_i:\, u(x, t)\ge k_i\} . Then, by using (4.5), we get:

    \begin{equation} \begin{aligned} I&\le \tilde\gamma \int_{\theta_i}^t \frac{1}{\tau}\tau\|u(\tau)\|_{L^{\infty}(B_R)}^{\sigma-1}\int_{B_R}u\left[G_{k_{i+1}}(u)\right]^{q-1}\,d\mu d\tau\\ & = \tilde\gamma \int_{\theta_i}^t \frac{1}{\tau}\tau\|u(\tau)\|_{L^{\infty}(B_R)}^{\sigma-1} \int_{B_R}k_i\frac{u}{k_i}\left[G_{k_{i+1}}(u)\right]^{q-1}\,d\mu d\tau\\ &\le \tilde\gamma \frac{k_i}{k_i-k_{i+1}}S(t)\int_{\theta_i}^{t}\frac{1}{\tau} \int_{B_R}\left[G_{k_{i+1}}(u)\right]^{q}\,d\mu d\tau. \end{aligned} \end{equation} (4.13)

    By substituting (4.13) into (4.12) we obtain

    \begin{equation*} \label{eq313} \begin{aligned} \sup\limits_{\tau_1 < \tau < t} &\int_{B_R}[G_{k_0}(u(x,t))]^{q}\,d\mu + \left(\frac p{p+q-2}\right)^p\iint_{U_{i-1}}\left |\nabla [G_{k_i}(u)]^{\frac{p+q-2}{p}} \right|^{p}\,d\mu d\tau\\ &\le \frac{2^i\,\tilde\gamma }{\tau_1-\tau_2}\iint_{U_i}[G_{k_{i+1}}(u)]^{q}\,d\mu d\tau+\frac{k_i\,\tilde\gamma}{k_i-k_{i+1}}\frac{S(t)}{\theta_0} \iint_{U_i} [G_{k_{i+1}}(u)]^{q}d\mu d\tau. \end{aligned} \end{equation*}

    To proceed further, observe that

    \frac{k_i}{k_i-k_{i+1}} = \frac{2^{i+1}a_2}{a_1-a_2}+2, \quad{\text{and}}\quad \theta_0\equiv\tau_1.

    Consequently, by choosing C_1 as in (4.6), we get

    \begin{equation*} \label{eq314} \begin{aligned} \sup\limits_{\tau_1 < \tau < t} &\int_{B_R}[G_{k_0}(u(x,t))]^{q}\,d\mu + \left(\frac p{p+q-2}\right)^p\iint_{U_{i-1}} |\nabla [G_{k_i}(u)]^{\frac{p+q-2}{p}} |^{p}\,d\mu d\tau\\ &\le 2^i\,\tilde\gamma\, C_1\int\int_{U_i}[G_{k_{i+1}}(u)]^{q}\,d\mu d\tau. \end{aligned} \end{equation*}

    The thesis follows, letting

    \begin{equation} \gamma: = \left[\min\left\{1;\,\left(\frac{p}{p+q-2}\right)^p\right\}\right]^{-1}\tilde\gamma. \end{equation} (4.14)

    Lemma 4.2. Assume (1.2), let 1 < r < q and assume that (1.3) holds. Let k_i , \theta_i , U_i be defined in (4.2), (4.3) and R > 0 . Let u be a solution to problem (3.2). Then, for every i\in\mathbb{N} and \varepsilon > 0 , we have

    \begin{equation*} \label{eq315} \begin{aligned} \sup\limits_{\tau_1 < \tau < t}&\int_{B_R}[G_{k_0}(u)]^q\,d\mu + \iint_{U_{i-1}}\left|\nabla[G_{k_i}(u)]^{\frac{p+q-2}{p}}\right|^p\,d\mu d\tau\\ &\le \varepsilon \iint_{U_i} \left|\nabla[G_{k_{i+1}}(u)]^{\frac{p+q-2}{p}}\right|^p \,d\mu d\tau\\ &+ C(\varepsilon)(2^i\gamma C_1)^{\frac{N(p+q-2-r)+pr}{N(p-2)+pr}}(t-\tau_2)\left(\sup\limits_{\tau_2 < \tau < t}\int_{B_R}[G_{k_{\infty}}(u)]^r\,d\mu\right)^{\frac{N(p-2)+pq}{N(p-2)+pr}}, \end{aligned} \end{equation*}

    with C_1 and \gamma defined as in (4.6) and (4.14) respectively and for some C(\varepsilon) > 0 .

    Proof. Let us fix q > 1 and 1 < r < q . We define

    \begin{equation} \alpha: = r\,\frac{N(p-2)+pq}{N(p+q-2-r)+pr}. \end{equation} (4.15)

    Observe that, since 1 < r < q , one has 0 < \alpha < q . By Hölder inequality with exponents \frac{pN}{N-p}\left(\frac{p+q-2}{p(q-\alpha)}\right) and \frac{N(p+q-2)}{N(p+\alpha-2)+p(q-\alpha)} , we thus have:

    \begin{equation} \begin{aligned} \int_{B_R}[G_{k_{i+1}}(u)]^q\,d\mu& = \int_{B_R}[G_{k_{i+1}}(u)]^{q-\alpha}[G_{k_{i+1}}(u)]^\alpha\,d\mu\\ &\le\left(\int_{B_R}[G_{k_{i+1}}(u)]^{\left(\frac{p+q-2}{p}\;\right)\frac{pN}{N-p}}\,d\mu\right)^{\left(\frac{p(q-\alpha)}{p+q-2}\;\right)\frac{N-p}{pN}}\\ &\quad\times \left(\int_{B_R}[G_{k_{i+1}}(u)]^{\frac{\alpha N(p+q-2)}{N(p+\alpha-2)+p(q-\alpha)}}\,d\mu\;\;\right)^{\frac{N(p+\alpha-2)+p(q-\alpha)}{N(p+q-2)}}\\ &\le \left(\left\|[G_{k_{i+1}}(u)]^{\frac{p+q-2}{p}}\right\|_{L^{p^*}(B_R)}\right)^{\frac{p(q-\alpha)}{p+q-2}} \\ &\quad\times \left(\int_{B_R}[G_{k_{i+1}}(u)]^{\frac{\alpha N(p+q-2)}{N(p+\alpha-2)+p(q-\alpha)}}\,d\mu\;\;\right)^{\frac{N(p+\alpha-2)+p(q-\alpha)}{N(p+q-2)}}. \end{aligned} \end{equation} (4.16)

    By the definition of \alpha in (4.15) and inequality (1.3), (4.16) becomes

    \begin{equation} \begin{aligned} \int_{B_R}[G_{k_{i+1}}(u)]^q\,d\mu\le\left(\frac{1}{C_{s,p}}\left\|\nabla[G_{k_{i+1}}(u)]^{\frac{p+q-2}{p}}\right\|_{L^{p}(B_R)}\right)^{\frac{p(q-\alpha)}{p+q-2}} \left(\int_{B_R}[G_{k_{i+1}}(u)]^{r}\,d\mu\right)^{\frac\alpha r}. \end{aligned} \end{equation} (4.17)

    We multiply both sides of (4.17) by 2^i \gamma C_1 with C_1 and \gamma as in (4.6) and (4.14), respectively. Then, we apply Young's inequality with exponents \frac{p+q-2}{q-\alpha} and \frac{p+q-2}{p+\alpha-2} to get:

    \begin{equation} \begin{aligned} &2^i\,\gamma C_1\int_{B_R}[G_{k_{i+1}}(u)]^q\,d\mu\\ &\le \varepsilon\int_{B_R}\left|\nabla[G_{k_{i+1}}(u)]^{\frac{p+q-2}{p}}\right|^p\,d\mu+C(\varepsilon)(2^i\gamma C_1)^{\frac{p+q-2}{p+\alpha-2}}\left(\int_{B_R}[G_{k_{i+1}}(u)]^{r}\,d\mu\right)^{\frac\alpha r\frac{p+q-2}{p+\alpha-2}} \end{aligned} \end{equation} (4.18)

    Define

    \lambda: = \frac\alpha r\left(\frac{p+q-2}{p+\alpha-2}\right) = \frac{N(p-2)+pq}{N(p-2)+pr}.

    Observe that \lambda > 1 since r < q . By Lemma 4.1,

    \begin{equation} \begin{aligned} \sup\limits_{\tau_1 < \tau < t}\int_{B_R}[G_{k_0}(u)]^q\,d\mu + \iint_{U_{i-1}}\left|\nabla[G_{k_i}(u)]^{\frac{p+q-2}{p}}\right|^p\,d\mu d\tau\le 2^i\gamma C_1\int_{\theta_i}^{t}&\int_{B_R}[G_{k_{i+1}}(u)]^q\,d\mu d\tau \end{aligned} \end{equation} (4.19)

    Moreover, let us integrate inequality (4.18) in the time interval \tau\in(\theta_i, t) . Then, we observe that

    \begin{equation} \begin{aligned} C(\varepsilon)(2^i\gamma C_1)^{\frac{p+q-2}{p+\alpha-2}}& \int_{\theta_i}^{t}\left(\int_{B_R}[G_{k_{i+1}}(u)]^{r}\,d\mu\right)^{\lambda}\,d\tau\\ &\le C(\varepsilon)(2^i\gamma C_1)^{\frac{p+q-2}{p+\alpha-2}}(t-\tau_2)\left(\sup\limits_{\tau_2 < \tau < t}\int_{B_R}[G_{k_{i+1}}(u)]^r\,d\mu\right)^\lambda \end{aligned} \end{equation} (4.20)

    where we have used that \tau_2 < \theta_i for every i\in\mathbb{N} . Finally, we substitute (4.19) and (4.20) into (4.18), thus we get

    \begin{aligned} \sup\limits_{\tau_1 < \tau < t}&\int_{B_R}[G_{k_0}(u)]^q\,d\mu + \iint_{U_{i-1}}\left|\nabla[G_{k_i}(u)]^{\frac{p+q-2}{p}}\right|^p\,d\mu d\tau\\ &\le \varepsilon\iint_{U_i}\left|\nabla[G_{k_{i+1}}(u)]^{\frac{p+q-2}{p}}\right|^p\,d\mu d\tau+C(\varepsilon)(2^i\gamma C_1)^{\frac{p+q-2}{p+\alpha-2}}(t-\tau_2)\left(\sup\limits_{\tau_2 < \tau < t}\int_{B_R}[G_{k_{i+1}}(u)]^r\,d\mu\right)^\lambda \end{aligned}

    The thesis follows by noticing that, for any i\in\mathbb{N}

    G_{k_i}(u)\le G_{k_{i+1}}(u)\le \ldots \le G_{k_\infty}(u),

    and that

    \frac{p+q-2}{p+\alpha-2} = \frac{N(p+q-2-r)+pr}{N(p-2)+pr}.

    Proposition 4.3. Assume that (1.2) and (1.3) holds. Let S(t) be defined as in (4.5). Let u be a solution to problem (3.2). Suppose that, for all t\in(0, T) ,

    S(t)\le 1.

    Let r\ge1 , then there exists k = k(p, r) such that

    \begin{equation*} \label{eq322} \|u(x,\tau)\|_{L^{\infty}\left(B_R\times\left(\frac t2,t\right)\right)}\,\le\, k \, t^{-\frac{N}{N(p-2)+pr}}\;\left[\sup\limits_{\frac t4 < \tau < t}\int_{B_R}u^r\,d\mu\right]^{\frac{p}{N(p-2)+pr}}, \end{equation*}

    for all t\in(0, T) .

    Proof. Let us define, for any j\in\mathbb{N} ,

    \begin{equation} J_i: = \iint_{U_{i}}\left|\nabla\left[G_{k_{i+1}}(u)\right]^{\frac{p+q-2}{p}}\right|^p\,d\mu\,dt, \end{equation} (4.21)

    where G_k , \{k_i\}_{i\in\mathbb{N}} and U_i have been defined in (4.1), (4.2) and (4.3) respectively. Let us fix 1\le r < q and define

    \begin{equation*} \label{eq324} \beta: = \frac{N(p+q-2-r)+pr}{N(p-2)+pr}. \end{equation*}

    By means of Lemma 4.2 and (4.21), we can write, for any i\in\mathbb{N}\cup\{0\}

    \begin{equation} \begin{aligned} \sup\limits_{\tau_1 < \tau < t}&\int_{B_R}[G_{k_0}(u)]^q\,d\mu + J_0\\ &\le \varepsilon J_1+ C(\varepsilon)(2\gamma C_1)^{\beta}(t-\tau_2)\left(\sup\limits_{\tau_2 < \tau < t}\int_{B_R}[G_{k_{\infty}}(u)]^r\,d\mu\right)^{\frac{N(p-2)+pq}{N(p-2)+pr}}\\ &\le \varepsilon \left\{\varepsilon J_2+ C(\varepsilon)(2^2\gamma C_1)^{\beta}(t-\tau_2)\left(\sup\limits_{\tau_2 < \tau < t}\int_{B_R}[G_{k_{\infty}}(u)]^r\,d\mu\right)^{\frac{N(p-2)+pq}{N(p-2)+pr}}\;\right\} \\ &\quad\quad+ C(\varepsilon)(2 \gamma C_1)^{\beta}(t-\tau_2)\left(\sup\limits_{\tau_2 < \tau < t}\int_{B_R}[G_{k_{\infty}}(u)]^r\,d\mu\right)^{\frac{N(p-2)+pq}{N(p-2)+pr}}\\ &\le \ldots\\ &\le \varepsilon^{i} J_{i}+\sum\limits_{j = 0}^{i-1}(2^{\beta}\varepsilon)^{j}(2 \gamma C_1)^\beta\,C(\varepsilon)(t-\tau_2)\left(\sup\limits_{\tau_2 < \tau < t}\int_{B_R}[G_{k_{\infty}}(u)]^r\,d\mu\right)^{\frac{N(p-2)+pq}{N(p-2)+pr}}. \end{aligned} \end{equation} (4.22)

    Fix now \varepsilon > 0 such that \varepsilon 2^\beta < \frac 12 . Taking the limit as i\longrightarrow+\infty in (4.22) we have:

    \begin{align} \sup\limits_{\tau_1 < \tau < t}\int_{B_R}[G_{k_0}(u)]^q\,d\mu\,\le\,\tilde C(2 \gamma C_1)^{\beta}(t-\tau_2)\left(\sup\limits_{\tau_2 < \tau < t}\int_{B_R}[G_{k_{\infty}}(u)]^r\,d\mu\right)^{\frac{N(p-2)+pq}{N(p-2)+pr}}. \end{align} (4.23)

    Observe that, due to the definition of the sequence \{k_i\}_{i\in\mathbb{N}} in (4.2), one has

    \begin{aligned} &k_0 = a_1\,,\quad\quad \quad \quad\quad \quad k_{\infty} = a_2\,;\\ &G_{k_0}(u) = G_{a_1}(u)\,, \quad\quad G_{k_\infty}(u) = G_{a_2}(u)\,. \end{aligned}

    For n\in\mathbb{N}\cup\{0\} , consider, for some C_0 > 0 to be fixed later, the following sequences

    \begin{equation} \begin{aligned} &t_n = \frac 12 t(1-2^{-n-1})\,;\\ &h_n = C_0(1-2^{-n-1})\,;\\ &\overline{h}_n = \frac 12(h_n+h_{n+1})\,. \end{aligned} \end{equation} (4.24)

    Let us now set in (4.23):

    \begin{equation} \tau_1 = t_{n+1}\,;\quad \tau_2 = t_n\,;\quad a_1 = \overline h_n\,;\quad a_2 = h_n\,. \end{equation} (4.25)

    Then the coefficient C_1 defined in (4.6), by (4.24) and (4.25), satisfies, since for any t\in(0, T) one has S(t)\le1 ,

    2C_1\le \frac{C_2^n}{t}\quad {\text{for some}}\,\,C_2 > 1.

    Due to the latter bound and to (4.25), (4.23) reads

    \begin{align} \sup\limits_{t_{n+1} < \tau < t}\int_{B_R}[G_{\overline h_n}(u)]^q\,d\mu\,\le\,\tilde C\,\gamma \,C_2^{n\beta}t^{-\beta+1}\left(\sup\limits_{t_n < \tau < t}\int_{B_R}[G_{h_{n}}(u)]^r\,d\mu\right)^{\frac{N(p-2)+pq}{N(p-2)+pr}}. \end{align} (4.26)

    Furthermore, observe that

    \begin{align} \int_{B_R}[G_{h_{n+1}}(u)]^r\,d\mu\,\le(h_{n+1}-\overline h_n)^{r-q}\int_{B_R}\left[G_{\overline h_n}(u)\right]^q\,d\mu. \end{align} (4.27)

    By combining together (4.26) and (4.27), we derive the following inequalities:

    \begin{equation} \begin{aligned} \sup\limits_{t_{n+1} < \tau < t}\int_{B_R}&[G_{h_{n+1}}(u)]^r\,d\mu\,\le (h_{n+1}-\overline h_n)^{r-q}\sup\limits_{t_{n+1} < \tau < t}\int_{B_R}\left[G_{\overline h_n}(u)\right]^q\,d\mu\\ &\le \tilde C\,\gamma\,C_2^{n\beta}\left(\frac{h_{n+1}-h_n}{2}\right)^{r-q}t^{-\beta+1}\left(\sup\limits_{t_n < \tau < t}\int_{B_R}[G_{h_{n}}(u)]^r\,d\mu\right)^{\frac{N(p-2)+pq}{N(p-2)+pr}}. \end{aligned} \end{equation} (4.28)

    Let us finally define

    Y_n: = \sup\limits_{t_n < \tau < t}\int_{B_R}[G_{h_{n}}(u)]^r\,d\mu.

    Hence, by using (4.24), (4.28) reads,

    \begin{equation*} \begin{aligned} Y_{n+1}&\le \tilde C\,\gamma\, C_2^{n\beta}\left(\frac{h_{n+1}-h_n}{2}\right)^{r-q}\,t^{-\beta+1}\,Y_n^{\frac{N(p-2)+pq}{N(p-2)+pr}}\\ &\le \tilde C\,\gamma\,C_2^{n\beta}\,2^{(n+3)(q-r)}\,C_0^{r-q}\,t^{-\beta+1}\,Y_n^{\frac{N(p-2)+pq}{N(p-2)+pr}}\,\\ &\le k^{n(q-r)}\,C_0^{r-q}\,t^{-\beta+1}\,Y_n^{\frac{N(p-2)+pq}{N(p-2)+pr}}\,, \end{aligned} \end{equation*}

    for some k = k(p, r) > 1 . From [25,Chapter 2,Lemma 5.6] it follows that

    \begin{equation} Y_n\longrightarrow 0\quad{\text{as}}\,\,\,n\to+\infty, \end{equation} (4.29)

    provided

    \begin{equation} C_0^{r-q}\,t^{-\beta+1}\,Y_0^{\frac{N(p-2)+pq}{N(p-2)+pr}-1}\le k^{r-q}. \end{equation} (4.30)

    Now, (4.29), in turn, reads

    \|u\|_{L^{\infty}\left(B_R\times\left(\frac t2,t\right)\right)}\,\le\, C_0.

    Moreover, (4.30) is fulfilled since

    C_0 = k\,t^{\frac{-\beta+1}{q-r}}\,Y_0^{\left(\frac{N(p-2)+pq}{N(p-2)+pr}-1\right)\left(\frac1{q-r}\right)}\le k\,t^{-\frac{N}{N(p-2)+pr}}\;\left[\sup\limits_{\frac t4 < \tau < t}\int_{B_R}u^r\,d\mu\right]^{\frac{p}{N(p-2)+pr}}.

    This concludes the proof.

    By Lemma 4.3, using the same arguments as in the proof of [27,Lemmata 4 and 5,and subsequent remarks], we get the following result.

    Lemma 5.1. Assume (1.2) and \sigma > p-1+\frac pN . Suppose that (1.3) and (2.2) hold. Let S(t) be defined as in (4.5). Define

    \begin{equation} T: = \sup\left\{t > 0:\,S(t)\le\,1\right\}. \end{equation} (5.1)

    Then

    \begin{equation*} \label{eq336} T = +\infty. \end{equation*}

    Proof of Theorem 2.2. Let \{u_{0, h}\}_{h\ge 0} be a sequence of functions such that

    \begin{equation*} \begin{aligned} &(a)\,\,u_{0,h}\in L^{\infty}(M)\cap C_c^{\infty}(M) \,\,\,{\text{for all}} \,\,h\ge 0, \\ &(b)\,\,u_{0,h}\ge 0 \,\,\,{\text{for all}} \,\,h\ge 0, \\ &(c)\,\,u_{0, h_1}\leq u_{0, h_2}\,\,\,{\text{for any }} h_1 < h_2, \\ &(d)\,\,u_{0,h}\longrightarrow u_0 \,\,\, {\text{in}}\,\, L^{s}(M)\cap L^1(M)\quad {\rm{ as }}\, h\to +\infty\,,\\ \end{aligned} \end{equation*}

    Observe that, due to assumptions (c) and (d) , u_{0, h} satisfies (2.2). For any R > 0 , k > 0 , h > 0 , consider the problem

    \begin{equation} \begin{cases} u_t = {{\rm{div}}}\left(|\nabla u|^{p-2}\nabla u\right) +T_k(u^{\sigma}) &{\text{in}}\,\, B_R\times (0,+\infty)\\ u = 0& {\text{in}}\,\, \partial B_R\times (0,\infty)\\ u = u_{0,h} &{\text{in}}\,\, B_R\times \{0\}\,. \\ \end{cases} \end{equation} (5.2)

    From standard results it follows that problem (5.2) has a solution u_{h, k}^R in the sense of Definition 3.1. In addition, u^R_{h, k}\in C\big([0, T]; L^q(B_R)\big) for any q > 1 .

    (ⅰ) In view of Proposition 4.3 and Lemma 5.1, the solution u_{h, k}^R to problem (5.2) satisfies estimate (4.3) for any t\in(0, +\infty) , uniformly w.r.t. R , k and h . By standard arguments we can pass to the limit as R\to\infty , k\to\infty and h\to\infty and we obtain a solution u to Eq (1.1) satisfying (2.3).

    (ⅱ) Due to Proposition 3.3, the solution u_{h, k}^R to problem (5.2) satisfies estimate (3.11) for any t\in(0, +\infty) , uniformly w.r.t. R , k and h . Thus, the solution u fulfills (2.5).

    (ⅲ) We now furthermore suppose that u_{0, h}\in L^q(M) and u_{0, h}\longrightarrow u_0 in L^{q}(M) . Due to Proposition 3.2, the solution u_{h, k}^R to problem (5.2) satisfies estimate (3.4) for any t\in(0, +\infty) , uniformly w.r.t. R , k and h . Thus, the solution u also fulfills (2.7).

    This completes the proof.

    To prove Theorem 2.4 we need the following two results.

    Lemma 6.1. Assume (1.2) and, moreover, that \sigma > p-1+\frac pN . Assume that inequality (1.3) holds. Let u be a solution of problem (3.2) with u_0\in L^{\infty}(B_R) , u_0\ge0 , such that

    \|u_0\|_{L^{\sigma_0}(B_R)}\le\varepsilon_2,

    for \varepsilon_2 = \varepsilon_2(\sigma, p, N, C_{s, p}, \sigma_0) > 0 sufficiently small and \sigma_0 as in (2.1). Let S(t) and T be defined as in (4.5) and (5.1) respectively. Then

    \begin{equation*} \label{eq335old} T = +\infty. \end{equation*}

    Proof. We suppose by contradiction that T < +\infty . Then, by (5.1) and (4.5), we can write:

    \begin{equation} 1 = S(T) = \sup\limits_{0 < t < T}\,t\|u(t)\|_{L^{\infty}(B_R)}^{\sigma-1}. \end{equation} (6.1)

    Due to Lemma 4.3 with the choice r = q > \sigma_0 , (6.1) reduces to

    \begin{equation} \begin{aligned} 1 = S(T)&\le \,\sup\limits_{0 < t < T}\,t\left\{k\,t^{-\frac{N}{N(p-2)+pq}}\;\left(\sup\limits_{\frac t4 < \tau < t}\int_{B_R}u^q\,d\mu\right)^{\frac{p}{N(p-2)+pq}}\;\right\}^{(\sigma-1)}\\ &\le \,\sup\limits_{0 < t < T}\,k\,t^{1-\frac{N(\sigma-1)}{N(p-2)+pq}}\;\left(\sup\limits_{\frac t4 < \tau < t}\left\|u(\tau)\right\|_{L^{q}(B_R)}^{\frac{q\,p(\sigma-1)}{N(p-2)+pq}}\;\right)\,. \end{aligned} \end{equation} (6.2)

    Define

    \begin{equation} I_1: = \sup\limits_{\frac t4 < \tau < t}\left\|u(\tau)\right\|_{L^{q}(B_R)}^{\frac{pq(\sigma-1)}{N(p-2)+pq}}. \end{equation} (6.3)

    In view of the choice q > \sigma_0 , we can apply Proposition 3.3 with q_0 = \sigma_0 to (6.3), thus we get

    \begin{equation} \begin{aligned} I_1&\le \sup\limits_{\frac t4 < \tau < t}\left[C\,t^{-\gamma_q}\,\|u_0\|^{\delta_q}_{L^{q_0}(B_R)}\right]^{\frac{pq(\sigma-1)}{N(p-2)+pq}}\\ & \le \,C\,t^{-\gamma_q\frac{pq(\sigma-1)}{N(p-2)+pq}}\;\;\,\|u_0\|^{\delta_q\frac{pq(\sigma-1)}{N(p-2)+pq}}_{L^{q_0}(B_R)}\,, \end{aligned} \end{equation} (6.4)

    where \gamma_q and \delta_q are defined in (3.12). By substituting (6.4) into (6.2) we get

    \begin{equation*} \label{eq340old} 1 = S(T)\le \,C\,k\sup\limits_{0 < t < T}\, t^{1-\frac{N(\sigma-1)}{N(p-2)+pq}-\gamma_q\frac{pq(\sigma-1)}{N(p-2)+pq}}\;\;\|u_0\|^{\delta_q\frac{pq(\sigma-1)}{N(p-2)+pq}}_{L^{q_0}(B_R)} \,. \end{equation*}

    Observe that

    \begin{aligned} &1-\frac{N(\sigma-1)}{N(p-2)+pq}-\gamma_q\frac{pq(\sigma-1)}{N(p-2)+pq} = 0;\\ & \delta_q\frac{pq(\sigma-1)}{N(p-2)+pq} = \sigma-p+1 > 0\,; \end{aligned}

    hence

    1 = S(T) < C\,\tilde C\, \varepsilon_2^{\sigma-p+1}\,.

    Provided \varepsilon_2 is sufficiently small, a contradiction, i.e., 1 = S(T) < 1 . Thus T = +\infty .

    Proposition 6.2. Assume (1.2) and, moreover, that \sigma > p-1+\frac pN . Let u be the solution to problem (3.2) with u_0\in{ \rm L}^{\infty}(B_R) , u_0\ge0 . Let \sigma_0 be defined in (2.1) and q > \sigma_0. Assume that

    \|u_0\|_{ \rm L^{\sigma_0}(B_R)}\, < \,\varepsilon_2

    with \varepsilon_2 = \varepsilon_2(\sigma, p, N, C_{s, p}, \sigma_0) > 0 sufficiently small. Then, for some C = C(N, \sigma, p, q, \sigma_0) > 0 :

    \begin{equation} \|u(t)\|_{L^{\infty}(B_R)}\le C\, t^{-\frac{1}{\sigma-1}}\,\|u_0\|_{L^{\sigma_0}(B_R)}^{1-\frac{p-2}{\sigma-1}}\,\quad for \;any\; t\in(0,+\infty) . \end{equation} (6.5)

    Proof. Due to Lemma 6.1,

    S(t)\le 1\quad {\text{for all}}\,\,\,t\in(0,+\infty).

    Therefore, by Lemma 4.3 and Proposition 3.3 with q_0 = \sigma_0 , for all t\in (0, +\infty)

    \begin{equation} \begin{aligned} \|u(t)\|_{L^{\infty}(B_R)}&\le \|u\|_{L^{\infty}\left(B_R\times\left(\frac t2,t\right)\right)}\,\\ &\le\,k\,t^{-\frac{N}{N(p-2)+pq}}\;\left[\sup\limits_{\frac t4 < \tau < t}\|u(\tau)\|_{L^q(B_R)}^q\right]^{\frac{p}{N(p-2)+pq}}\\ &\le\,C\,t^{-\frac{N}{N(p-2)+pq}-\gamma_q\frac{pq}{N(p-2)+pq}}\;\|u_0\|_{L^{\sigma_0}(B_R)}^{\delta_q\frac{pq}{N(p-2)+pq}}\,, \end{aligned} \end{equation} (6.6)

    where C = C(\sigma, p, N, q, \sigma_0) > 0 , \gamma_q and \delta_q as in (3.12) with q_0 = \sigma_0 . Observe that

    \begin{equation} -\frac{N}{N(p-2)+pq}-\gamma_q\frac{pq}{N(p-2)+pq} = -\frac 1{\sigma-1}\,, \end{equation} (6.7)

    and

    \begin{equation} \delta_q\frac{pq}{N(p-2)+pq} = \frac{\sigma-p+1}{\sigma-1}\,. \end{equation} (6.8)

    By combining (6.6) with (6.7) and (6.8) we get the thesis.

    Proof of Theorem 2.4. We use the same argument discussed in the proof of Theorem 2.2. In fact, let \{u_{0, l}\}_{l\ge 0} be a sequence of functions such that

    \begin{equation*} \begin{aligned} &(a)\,\,u_{0,l}\in L^{\infty}(M)\cap C_c^{\infty}(M) \,\,\,{\text{for all}} \,\,l\ge 0, \\ &(b)\,\,u_{0,l}\ge 0 \,\,\,{\text{for all}} \,\,l\ge 0, \\ &(c)\,\,u_{0, l_1}\leq u_{0, l_2}\,\,\,{\text{for any }} l_1 < l_2, \\ &(d)\,\,u_{0,l}\longrightarrow u_0 \,\,\, {\text{in}}\,\, L^{\sigma_0}(M)\quad {\rm{ as }}\, l\to +\infty\,,\\ \end{aligned} \end{equation*}

    where \sigma_0 has been defined in (2.1). Observe that, due to assumptions (c) and (d) , u_{0, l} satisfies (2.10). For any R > 0 , k > 0 , l > 0 , we consider problem (5.2) with the sequence u_{0, h} replaced by the sequence u_{0, l} . From standard results it follows that problem (5.2) has a solution u_{l, k}^R in the sense of Definition 3.1; moreover, u^R_{l, k}\in C\big([0, T]; L^q(B_R)\big) for any q > 1 .

    Due to Proposition 6.2, Proposition 3.3 and Lemma 3.2, the solution u_{l, k}^R to problem (5.2) satisfies estimates (6.5), (3.11) and (3.4) for t\in(0, +\infty) , uniformly w.r.t. R , k and l . Thus, by standard arguments we can pass to the limit as R\to\infty , k\to\infty and l\to\infty and we obtain a solution u to Eq (1.1) satisfying (2.11), (2.5) and (2.7).

    Lemma 7.1. Assume (1.2), p > 2 , and q > \max\left\{\sigma_0, 1\right\} . Let u be a solution to problem (3.2) with u_0\in L^{\infty}(B_R) , u_0\ge0 , such that

    \begin{equation} \|u_0\|_{L^{q}(B_R)}\le\delta_1, \end{equation} (7.1)

    for \delta_1 > 0 sufficiently small. Let S(t) be as in (4.5), then

    \begin{equation} T: = \sup\{t > 0:\,S(t)\le\,1\} > 1. \end{equation} (7.2)

    Proof. By (4.5) and (7.2) one has

    \begin{equation} 1 = S(T) = \sup\limits_{0 < t < T}\,t\|u(t)\|_{L^{\infty}(B_R)}^{\sigma-1}. \end{equation} (7.3)

    By Lemma (4.3) applied with r = q > \max\left\{\frac Np(\sigma-p+1), 1\right\} , (7.3) gives

    \begin{equation} \begin{aligned} 1 = S(T)&\le \sup\limits_{0 < t < T}\,t\left\{k\,t^{-\frac{N}{N(p-2)+pq}}\;\left(\sup\limits_{\frac t4 < \tau < t}\int_{B_R}u^q\,d\mu\right)^{\frac{p}{N(p-2)+pq}}\right\}^{(\sigma-1)}\\ &\le \sup\limits_{0 < t < T}\, k\,t^{1-\frac{N(\sigma-1)}{N(p-2)+pq}}\;\left(\sup\limits_{\frac t4 < \tau < t}\left\|u(\tau)\right\|_{L^{q}(B_R)}^{\frac{q\,p(\sigma-1)}{N(p-2)+pq}}\right)\,. \end{aligned} \end{equation} (7.4)

    By applying Proposition 3.6 to (7.4) and due to (7.1), we get

    \begin{equation*} \label{eq65} \begin{aligned} 1 = S(T)&\le \sup\limits_{0 < t < T}\, k\,t^{1-\frac{N(\sigma-1)}{N(p-2)+pq}}\;\left\|u_0\right\|_{L^{q}(B_R)}^{\frac{q\,p(\sigma-1)}{N(p-2)+pq}}\\ &\le \, k\,T^{1-\frac{N(\sigma-1)}{N(p-2)+pq}}\;\;\,\delta_1^{\frac{q\,p(\sigma-1)}{N(p-2)+pq}}\,. \end{aligned} \end{equation*}

    The thesis follows for \delta_1 > 0 small enough.

    Lemma 7.2. Assume (1.2), p > 2 and s > \max\left\{\sigma_0, 1\right\}. Let u be a solution to problem (3.2) with u_0\in L^{\infty}(B_R) , u_0\ge0 , such that

    \begin{equation} \|u_0\|_{L^{s}(B_R)}\le\delta_1,\quad \|u_0\|_{L^{\sigma\frac Np}(B_R)}\le\delta_1, \end{equation} (7.5)

    for \delta_1 > 0 sufficiently small. Let S(t) be as in (4.5), then

    \begin{equation} T: = \sup\{t\ge0:\,S(t)\le\,1\} = +\infty. \end{equation} (7.6)

    Proof. We suppose by contradiction that

    T < +\infty.

    Then, by (7.6), the definition of S(t) in (4.5) and by Lemma 7.1 we can write,

    \begin{equation} \begin{aligned} 1 = S(T)& = \sup\limits_{0 < t < T}\,t\|u(t)\|_{L^{\infty}(B_R)}^{\sigma-1}\\ &\le \sup\limits_{0 < t < 1}\,t\|u(t)\|_{L^{\infty}(B_R)}^{\sigma-1}+ \sup\limits_{1 < t < T}\,t\|u(t)\|_{L^{\infty}(B_R)}^{\sigma-1}\\ & = :J_1+J_2\,. \end{aligned} \end{equation} (7.7)

    Now, by Lemma 4.3, applied with r = s , and Lemma 3.5 with q = s , we can write

    \begin{equation} \begin{aligned} J_1&\le \,\sup\limits_{0 < t < 1}\,t\left\{k\,t^{-\frac{N}{N(p-2)+ps}}\;\left(\sup\limits_{\frac t4 < \tau < t}\int_{B_R}u^s\,d\mu\right)^{\frac{p}{N(p-2)+ps}}\right\}^{(\sigma-1)}\\ &\le \,\sup\limits_{0 < t < 1}\, k\,t^{1-\frac{N(\sigma-1)}{N(p-2)+ps}}\;\left\|u_0\right\|_{L^{s}(B_R)}^{\frac{ps(\sigma-1)}{N(p-2)+ps}}\,. \end{aligned} \end{equation} (7.8)

    On the other hand, for any q > s , by Lemma 4.3, applied with r = q , and Proposition 3.6 with q_0 = s , we get

    \begin{equation} \begin{aligned} J_2&\le \,\sup\limits_{1 < t < T}\,t\left\{k\,t^{-\frac{N}{N(p-2)+pq}}\;\left(\sup\limits_{\frac t4 < \tau < t}\int_{B_R}u^q\,d\mu\right)^{\frac{p}{N(p-2)+pq}}\;\right\}^{(\sigma-1)}\\ &\le \,\sup\limits_{1 < t < T}\, k\,t^{1-\frac{N(\sigma-1)}{N(p-2)+pq}}\sup\limits_{\frac t4 < \tau < t}\left\|u(\tau)\right\|_{L^{q}(B_R)}^{\frac{pq(\sigma-1)}{N(p-2)+pq}}\\ &\le \,\sup\limits_{1 < t < T}\, k\,t^{1-\frac{N(\sigma-1)}{N(p-2)+pq}}\sup\limits_{\frac t4 < \tau < t}\left(Ct^{-\frac{s}{p-2}\left(\frac 1{s}-\frac 1q\right)}\left\|u_0\right\|_{L^{s}(B_R)}^{\frac{s}{q}}\right)^{\frac{pq(\sigma-1)}{N(p-2)+pq}}\\ &\le \,\sup\limits_{1 < t < T}\,\frac{C\, k}{4}\,t^{1-\frac{N(\sigma-1)}{N(p-2)+pq}-\frac{spq(\sigma-1)}{(p-2)[N(p-2)+pq]}\;\;\left(\frac 1{s}-\frac 1q\right)}\left\|u_0\right\|_{L^{s}(B_R)}^{\frac{ps(\sigma-1)}{N(p-2)+pq}}\,. \end{aligned} \end{equation} (7.9)

    By substituting (7.8) and (7.9) into (7.7) we get

    \begin{equation} 1 = S(T)\le \sup\limits_{0 < t < 1}\, k\,t^{a}\left\|u_0\right\|_{L^{s}(B_R)}^{\frac{ps(\sigma-1)}{N(p-2)+ps}}+\sup\limits_{1 < t < T}\,\frac{C\, k}{4}\,t^{b}\left\|u_0\right\|_{L^{s}(B_R)}^{\frac{ps(\sigma-1)}{N(p-2)+pq}}\,, \end{equation} (7.10)

    where we have set

    a = 1-\frac{N(\sigma-1)}{N(p-2)+ps},\quad{\text{and}}\quad b = 1-\frac{N(\sigma-1)}{N(p-2)+pq}-\frac{spq(\sigma-1)}{(p-2)[N(p-2)+pq]}\left(\frac 1{s}-\frac 1q\right)\,.

    Now, observe that, since s > \max\left\{\frac Np(\sigma-p+1), 1\right\} and q > s ,

    a > 0;\quad{\text{and}}\quad b < 0\,.

    Hence, (7.10), due to assumption (7.5), reads

    \begin{equation*} 1 = S(T) < k\, \delta_1^{\frac{ps(\sigma-1)}{N(p-2)+ps}}\,+\,\frac{C\, k}{4}\delta_1^{\frac{ps(\sigma-1)}{N(p-2)+pq}}\,. \end{equation*}

    Provided that \delta_1 is sufficiently small, thus yielding 1 = S(T) < 1 , a contradiction. Thus T = +\infty .

    Proposition 7.3. Assume (1.2), p > 2 and s > \max\left\{\sigma_0, 1\right\} . Let u be a solution to problem (3.2) with u_0\in L^{\infty}(B_R) , u_0\ge0 , such that

    \|u_0\|_{L^{s}(B_R)}\le\varepsilon_1,\quad \|u_0\|_{L^{\sigma\frac Np}(B_R)}\le\varepsilon_1,

    with \varepsilon_1 = \varepsilon_1(\sigma, p, N, C_{s, p}, C_p, s) sufficiently small. Then, for any t\in(0, +\infty) , for some \Gamma = \Gamma(\sigma, p, N, q, s, C_{s, p}, C_p) > 0

    \begin{equation} \|u(t)\|_{L^{\infty}(B_R)}\le \Gamma\, t^{-\frac{1}{p-2}\left(1-\frac{ps}{N(p-2)+pq}\right)}\,\|u_0\|_{L^{s}(B_R)}^{\frac{ps}{N(p-2)+pq}}\,. \end{equation} (7.11)

    Proof. Due to Lemma 7.2,

    S(t)\le 1\quad {\text{for all}}\,\,\,t\in(0,+\infty].

    Therefore, by Lemma 4.3 and Proposition 3.6 applied with q_0 = s , for any q > s , we get, for all t\in (0, +\infty)

    \begin{equation*} \begin{aligned} \|u(t)\|_{L^{\infty}(B_R)}&\le \|u\|_{L^{\infty}\left(B_R\times\left(\frac t2,t\right)\right)}\,\\ &\le\, k\,t^{-\frac{N}{N(p-2)+pq}}\;\left[\sup\limits_{\frac t4 < \tau < t}\|u(\tau)\|_{L^q(B_R)}^q\right]^{\frac{p}{N(p-2)+pq}}\\ &\le\,\Gamma\,t^{-\frac{N}{N(p-2)+pq}-\frac{s}{p-2}\left(\frac 1{s}-\frac 1q\right)\frac{pq}{N(p-2)+pq}}\;\|u_0\|_{L^{s}(B_R)}^{\frac{s}{q}\frac{pq}{N(p-2)+pq}}\,. \end{aligned} \end{equation*}

    Observing that

    \begin{equation*} -\frac{N}{N(p-2)+pq}-\frac{s}{p-2}\left(\frac 1{s}-\frac 1q\right)\frac{pq}{N(p-2)+pq} = -\frac 1{p-2}\left(1-\frac{ps}{N(p-2)+pq}\right)\,, \end{equation*}

    we get the thesis.

    Proof of Theorem 2.7. We proceed as in the proof of the previous Theorems. Let \{u_{0, h}\}_{h\ge 0} be a sequence of functions such that

    \begin{equation} \begin{aligned} &(a)\,\,u_{0,h}\in L^{\infty}(M)\cap C_c^{\infty}(M) \,\,\,{\text{for all}} \,\,h\ge 0, \\ &(b)\,\,u_{0,h}\ge 0 \,\,\,{\text{for all}} \,\,h\ge 0, \\ &(c)\,\,u_{0, h_1}\leq u_{0, h_2}\,\,\,{\text{for any }} h_1 < h_2, \\ &(d)\,\,u_{0,h}\longrightarrow u_0 \,\,\, {\text{in}}\,\, L^{s}(M)\quad {\rm{ as }}\, h\to +\infty\,.\\ \end{aligned} \end{equation} (7.12)

    From standard results it follows that problem (5.2) has a solution u_{h, k}^R in the sense of Definition 3.1 with u_{0, h} as in (7.12); moreover, u^R_{h, k}\in C\big([0, \infty); L^q(B_R)\big) for any q > 1 . Due to Proposition 7.3, 3.6 and Lemmata 3.5 and 7.2, the solution u_{h, k}^R to problem (5.2) satisfies estimates (3.31), (3.42) and (7.11) for any t\in(0, +\infty) , uniformly w.r.t. R , k and h . Thus, by standard arguments, we can pass to the limit as R\to+\infty , k\to+\infty and h\to+\infty and we obtain a solution u to problem (1.1), which fulfills (2.12), (2.13) and (2.14).

    We now consider the following nonlinear reaction-diffusion problem:

    \begin{equation} \begin{cases} \, u_t = \Delta u^m +\, u^{\sigma} & {\text{in}}\,\, M\times (0,T) \\ \,\; u = u_0 &{\text{in}}\,\, M\times \{0\}\,, \end{cases} \end{equation} (8.1)

    where M is an N- dimensional complete noncompact Riemannian manifold of infinite volume, \Delta being the Laplace-Beltrami operator on M and T\in (0, \infty] . We shall assume throughout this section that

    N\geq 3,\quad \quad m\, > \,1,\quad \quad \sigma\, > \,m,

    so that we are concerned with the case of degenerate diffusions of porous medium type (see [37]), and that the initial datum u_0 is nonnegative. Let L ^q(M) be the space of those measurable functions f such that |f|^q is integrable w.r.t. the Riemannian measure \mu . We shall always assume that M supports the Sobolev inequality, namely that:

    \begin{equation} ( {\rm{Sobolev\ inequality)}}\ \ \ \ \ \ \|v\|_{L^{2^*}(M)} \le \frac{1}{C_s} \|\nabla v\|_{L^2(M)}\quad {\text{for any}}\,\,\, v\in C_c^{\infty}(M), \end{equation} (8.2)

    where C_s is a positive constant and 2^*: = \frac{2N}{N-2} . In one of our main results, we shall also suppose that M supports the Poincaré inequality, namely that:

    \begin{equation} ( {\rm{Poincaré\ inequality)}}\ \ \ \ \ \|v\|_{L^2(M)} \le \frac{1}{C_p} \|\nabla v\|_{L^2(M)} \quad {\text{for any}}\,\,\, v\in C_c^{\infty}(M), \end{equation} (8.3)

    for some C_p > 0 .

    Solutions to (8.1) will be meant in the very weak, or distributional, sense, according to the following definition.

    Definition 8.1. Let M be a complete noncompact Riemannian manifold of infinite volume, of dimension N\ge3 . Let m > 1 , \sigma > m and u_0\in{ \rm L}^{1}_{loc}(M) , u_0\ge0 . We say that the function u is a solution to problem (8.1) in the time interval [0, T) if

    u\in L^{\sigma}_{loc}(M\times(0,T))

    and for any \varphi \in C_c^{\infty}(M\times[0, T]) such that \varphi(x, T) = 0 for any x\in M , u satisfies the equality:

    \begin{equation*} \begin{aligned} -\int_0^T\int_{M} \,u\,\varphi_t\,d\mu\,dt = &\int_0^T\int_{M} u^m\,\Delta\varphi\,d\mu\,dt\,+ \int_0^T\int_{M} \,u^{\sigma}\,\varphi\,d\mu\,dt \\ & +\int_{M} \,u_0(x)\,\varphi(x,0)\,d\mu. \end{aligned} \end{equation*}

    First we consider the case that \sigma > m+\frac 2 N and the Sobolev inequality holds on M . In order to state our results we define

    \begin{equation} \sigma_1: = (\sigma-m)\frac{N}{2}. \end{equation} (8.4)

    Observe that \sigma_1 > 1 whenever \sigma > m+\frac 2N . We comment that the next results improve and in part correct some of the results of [17]. The proofs are omitted since they are identical to the previous ones.

    Theorem 8.2. Let M be a complete, noncompact, Riemannian manifold of infinite volume and of dimension N\ge3 , such that the Sobolev inequality (8.2) holds. Let m > 1 , \sigma > m+\frac{2}{N} , s > \sigma_1 and u_0\in{ \rm L}^{s}(M)\cap L^1(M) , u_0\ge0 .

    (ⅰ) Assume that

    \begin{equation*} \label{a0} \|u_0\|_{ \rm L^{s}(M)}\, < \,\varepsilon_0,\quad \|u_0\|_{ \rm L^{1}(M)} < \,\varepsilon_0\,, \end{equation*}

    with \varepsilon_0 = \varepsilon_0(\sigma, m, N, C_{s}) > 0 sufficiently small. Then problem (8.1) admits a solution for any T > 0 , in the sense of Definition 8.1. Moreover, for any \tau > 0, one has u\in L^{\infty}(M\times(\tau, +\infty)) and there exists a constant \Gamma > 0 such that, one has

    \begin{equation*} \label{aeq21tot} \|u(t)\|_{L^{\infty}(M)}\le \Gamma\, t^{-\alpha}\,\|u_0\|_{L^{1}(M)}^{\frac{2}{N(m-1)+2}}\,\quad\mathit{{\text{for all $t > 0$,}}} \end{equation*}

    where

    \alpha: = \frac{N}{N(m-1)+2}\,.

    (ⅱ) Let \sigma_1\le q < \infty and

    \begin{equation*} \label{a2} \|u_0\|_{L^{\sigma_1}(M)} < \hat \varepsilon_0 \end{equation*}

    for \hat\varepsilon_0 = \hat\varepsilon_0(\sigma, m, N, C_s, q) > 0 small enough. Then there exists a constant C = C(m, \sigma, N, \varepsilon_0, C_s, q) > 0 such that

    \begin{equation*} \label{a3} \|u(t)\|_{L^q(M)}\le C\,t^{-\gamma_q} \|u_{0}\|^{\delta_q}_{L^{\sigma_1}(M)}\quad for\; all\,\, t > 0\,, \end{equation*}

    where

    \gamma_q = \frac{1}{\sigma-1}\left[1-\frac{N(\sigma-m)}{2q}\right],\quad \delta_q = \frac{\sigma-m}{\sigma-1}\left[1+\frac{N(m-1)}{2q}\right]\,.

    (ⅲ) Finally, for any 1 < q < \infty , if u_0\in { \rm L}^q(M)\cap \rm L^{\sigma_1}(M) and

    \begin{equation*} \label{a5} \|u_0\|_{ \rm L^{\sigma_1}(M)}\, < \,\varepsilon \end{equation*}

    with \varepsilon = \varepsilon(\sigma, m, N, r, C_s, q) > 0 sufficiently small, then

    \begin{equation*} \label{a6} \|u(t)\|_{L^q(M)}\le \|u_{0}\|_{L^q(M)}\quad for\; all\,\, t > 0\,. \end{equation*}

    Theorem 8.3. Let M be a complete, noncompact manifold of infinite volume and of dimension N\ge3 , such that the Sobolev inequality (8.2) holds. Let m > 1 , \sigma > m+\frac{2}{N} and u_0\in{ \rm L}^{\sigma_1}(M) , u_0\ge0 where \sigma_1 has been defined in (8.4). Assume that

    \begin{equation*} \label{a1} \|u_0\|_{ \rm L^{\sigma_1}(M)}\, < \,\varepsilon_0 \end{equation*}

    with \varepsilon_0 = \varepsilon_0(\sigma, m, N, r, C_s) > 0 sufficiently small. Then problem (8.1) admits a solution for any T > 0 , in the sense of Definition 8.1. Moreover, for any \tau > 0, one has u\in L^{\infty}(M\times(\tau, +\infty)) and there exists a constant \Gamma > 0 such that, one has

    \begin{equation*} \|u(t)\|_{L^{\infty}(M)}\le \Gamma\, t^{-\frac1{\sigma-1}}\|u_0\|_{L^{\sigma_1}(M)}^{\frac{\sigma-m}{\sigma-1}}\quad \mathit{{\text{for all $t > 0$.}}} \end{equation*}

    Moreover, the statements in (ⅱ) and (ⅲ) of Theorem 8.2 hold.

    In the next theorem, we address the case that \sigma > m , supposing that both the inequalities (8.2) and (8.3) hold on M .

    Theorem 8.4. Let M be a complete, noncompact manifold of infinite volume and of dimension N\ge3 , such that the Sobolev inequality (8.2) and the Poincaré inequality (8.3) hold. Let

    m > 1,\quad \sigma > m,

    and u_0\in{ \rm L}^{s}(M)\cap { \rm L}^{\sigma\frac N2}(M) where s > \max\left\{1, \sigma_1\right\} , u_0\ge0 . Assume that

    \begin{equation*} \label{a7} \left\| u_0\right\|_{L^{s}(M)}\, < \,\varepsilon_1, \quad \left\| u_0\right\|_{L^{\sigma\frac N2}(M)}\, < \,\varepsilon_1, \end{equation*}

    holds with \varepsilon_1 = \varepsilon_1(m, \sigma, N, r, C_p, C_s) > 0 sufficiently small. Then problem (8.1) admits a solution for any T > 0 , in the sense of Definition 8.1. Moreover for any \tau > 0 and for any q > s one has u\in L^{\infty}(M\times(\tau, +\infty)) and for all t > 0 one has

    \begin{equation*} \label{a8} \|u(t)\|_{L^{\infty}(B_R)}\le \Gamma\, t^{-\beta_{q,s}}\,\|u_0\|_{L^{s}(B_R)}^{\frac{2s}{N(m-1)+2q}}\,, \end{equation*}

    where

    \begin{equation*} \label{a9} \beta_{q,s}: = \frac{1}{m-1}\left(1-\frac{2s}{N(m-1)+2q}\right) > 0\,. \end{equation*}

    Moreover, let s\le q < \infty and

    \begin{equation*} \label{a10} \|u_0\|_{L^{s}(M)} < \hat\varepsilon_1, \end{equation*}

    for some \hat\varepsilon_1 = \hat \varepsilon_1(\sigma, m, N, r, C_p, C_s, q, s) > 0 sufficiently small. Then there exists a constant C = C(\sigma, m, N, \varepsilon_1, C_s, C_p, q, s) > 0 such that

    \begin{equation*} \label{a11} \|u(t)\|_{L^q(M)}\le Ct^{-\gamma_q} \|u_{0}\|_{L^s(M)}^{\delta_q}\quad for\; all \,\, t > 0\,, \end{equation*}

    where

    \gamma_q: = \frac{s}{m-1}\left[\frac 1s-\frac 1q\right],\quad\quad \delta_q: = \frac sq.

    Finally, for any 1 < q < \infty , if u_0\in L^q(M)\cap L^s(M)\cap { \rm L}^{\sigma\frac N2}(M) and

    \begin{equation*} \|u_0\|_{L^{s}(M)} < \varepsilon, \end{equation*}

    for some \varepsilon = \varepsilon(\sigma, m, N, C_p, C_s, q) > 0 sufficiently small, then

    \begin{equation*} \label{a12} \|u(t)\|_{L^q(M)}\le \|u_{0}\|_{L^q(M)}\quad for\; all\,\, t > 0\,. \end{equation*}

    The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA, Italy) of the Istituto Nazionale di Alta Matematica (INdAM, Italy) and are partially supported by the PRIN project 201758MTR2: "Direct and Inverse Problems for Partial Differential Equations: Theoretical Aspects and Applications" (Italy).

    The authors declare no conflict of interest.



    [1] E. Acerbi, G. Mingione, Gradient estimates for a class of parabolic systems, Duke Math. J., 136 (2007), 285–320. https://doi.org/10.1215/S0012-7094-07-13623-8 doi: 10.1215/S0012-7094-07-13623-8
    [2] N. D. Alikakos, L^p bounds of solutions of reaction-diffusion equations, Commun. Part. Diff. Eq., 4 (1979), 827–868. https://doi.org/10.1080/03605307908820113 doi: 10.1080/03605307908820113
    [3] C. Bandle, M. A. Pozio, A. Tesei, The Fujita exponent for the Cauchy problem in the hyperbolic space, J. Differ. Equations, 251 (2011), 2143–2163. https://doi.org/10.1016/j.jde.2011.06.001 doi: 10.1016/j.jde.2011.06.001
    [4] P. Bénilan, Opérateurs accrétifs et semi-groupes dans les espaces L^p (1 \le p \le +\infty), Japan-France seminar, Japan Society for the Advancement of Science, 1978.
    [5] P. Bénilan, M. G. Crandall, Completely accretive operators, semigroup theory and evolution equations (Delft, 1989), In: Lecture Notes in Pure and Applied Mathematics, Volume 135, Dekker, 1991, 41–75.
    [6] V. Bögelein, F. Duzaar, G. Mingione, The regularity of general parabolic systems with degenerate diffusion, Memoirs of the American Mathematical Society, 2013. https://doi.org/10.1090/S0065-9266-2012-00664-2
    [7] X. Chen, M. Fila, J. S. Guo, Boundedness of global solutions of a supercritical parabolic equation, Nonlinear Anal., 68 (2008), 621–628. https://doi.org/10.1016/j.na.2006.11.023 doi: 10.1016/j.na.2006.11.023
    [8] T. Coulhon, D. Hauer, Regularisation effects of nonlinear semigroups, SMAI - Mathématiques et Applications, Springer, to appear.
    [9] K. Deng, H. A. Levine, The role of critical exponents in blow-up theorems: the sequel, J. Math. Anal. Appl., 243 (2000), 85–126. https://doi.org/10.1006/jmaa.1999.6663 doi: 10.1006/jmaa.1999.6663
    [10] H. Fujita, On the blowing up of solutions of the Cauchy problem for u_t = \Delta u+u^{1+\alpha}, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109–124.
    [11] Y. Fujishima, K. Ishige, Blow-up set for type I blowing up solutions for a semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 231–247. https://doi.org/10.1016/j.anihpc.2013.03.001 doi: 10.1016/j.anihpc.2013.03.001
    [12] V. A. Galaktionov, The conditions for there to be no global solutions of a class of quasilinear parabolic equations, USSR Computational Mathematics and Mathematical Physics, 22 (1982), 73–90. https://doi.org/10.1016/0041-5553(82)90037-4 doi: 10.1016/0041-5553(82)90037-4
    [13] V. A. Galaktionov, Blow-up for quasilinear heat equations with critical Fujita's exponents, Proc. Roy. Soc. Edinb. A, 124 (1994), 517–525. https://doi.org/10.1017/S0308210500028766 doi: 10.1017/S0308210500028766
    [14] V. A. Galaktionov, H. A. Levine, A general approach to critical Fujita exponents in nonlinear parabolic problems, Nonlinear Anal., 34 (1998), 1005–1027. https://doi.org/10.1016/S0362-546X(97)00716-5 doi: 10.1016/S0362-546X(97)00716-5
    [15] A. Grigor'yan, Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc., 36 (1999), 135–249. https://doi.org/10.1090/s0273-0979-99-00776-4 doi: 10.1090/s0273-0979-99-00776-4
    [16] A. Grigor'yan, Heat kernel and analysis on manifolds, Providence, RI: American Mathematical Society, 2009.
    [17] G. Grillo, G. Meglioli, F. Punzo, Global existence of solutions and smoothing effects for classes of reaction-diffusion equations on manifolds, J. Evol. Equ., 21 (2021), 2339–2375. https://doi.org/10.1007/s00028-021-00685-3 doi: 10.1007/s00028-021-00685-3
    [18] G. Grillo, G. Meglioli, F. Punzo, Smoothing effects and infinite time blowup for reaction-diffusion equations: an approach via Sobolev and Poincaré inequalities, J. Math. Pure. Appl., 151 (2021), 99–131. https://doi.org/10.1016/j.matpur.2021.04.011 doi: 10.1016/j.matpur.2021.04.011
    [19] G. Grillo, G. Meglioli, F. Punzo, Blow-up versus global existence of solutions for reaction-diffusion equations on classes of Riemannian manifolds, Annali di Matematica Pura e Applicata, in press. https://doi.org/10.1007/s10231-022-01279-7
    [20] Q. Gu, Y. Sun, J. Xiao, F. Xu, Global positive solution to a semi-linear parabolic equation with potential on Riemannian manifold, Calc. Var., 59 (2020), 170. https://doi.org/10.1007/s00526-020-01837-y doi: 10.1007/s00526-020-01837-y
    [21] D. Hauer, Regularizing effect of homogeneous evolution equations with perturbation, Nonlinear Anal., 206 (2021), 112245. https://doi.org/10.1016/j.na.2021.112245 doi: 10.1016/j.na.2021.112245
    [22] K. Hayakawa, On nonexistence of global solutions of some semilinear parabolic differential equations, Proc. Japan Acad., 49 (1973), 503–505. https://doi.org/10.3792/pja/1195519254 doi: 10.3792/pja/1195519254
    [23] K. Kobayashi, T. Sirao, H. Tanaka, On the growing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407–424. https://doi.org/10.2969/jmsj/02930407 doi: 10.2969/jmsj/02930407
    [24] T. Kuusi, G. Mingione, Gradient regularity for nonlinear parabolic equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 12 (2013), 755–822. https://doi.org/10.2422/2036-2145.201103_006 doi: 10.2422/2036-2145.201103_006
    [25] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Ural'tseva, Linear and quasilinear equations of parabolic type, Providence, RI: American Mathematical Society, 1968.
    [26] H. A. Levine, The role of critical exponents in blow-up theorems, SIAM Rev., 32 (1990), 262–288. https://doi.org/10.1137/1032046 doi: 10.1137/1032046
    [27] A. V. Martynenko, A. F. Tedeev, On the behavior of solutions of the Cauchy problem for a degenerate parabolic equation with nonhomogeneous density and a source, Comput. Math. Math. Phys., 48 (2008), 1145–1160. https://doi.org/10.1134/S0965542508070087 doi: 10.1134/S0965542508070087
    [28] P. Mastrolia, D. D. Monticelli, F. Punzo, Nonexistence of solutions to parabolic differential inequalities with a potential on Riemannian manifolds, Math. Ann., 367 (2017), 929–963. https://doi.org/10.1007/s00208-016-1393-2 doi: 10.1007/s00208-016-1393-2
    [29] G. Meglioli, D. D. Monticelli, F. Punzo, Nonexistence of solutions to quasilinear parabolic equations with a potential in bounded domains, Calc. Var., 61 (2022), 23. https://doi.org/10.1007/s00526-021-02132-0 doi: 10.1007/s00526-021-02132-0
    [30] E. Mitidieri, S. I. Pohozaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Proc. Steklov Inst. Math., 234 (2001), 1–362.
    [31] E. Mitidieri, S. I. Pohozaev, Towards a unified approach to nonexistence of solutions for a class of differential inequalities, Milan J. Math., 72 (2004), 129–162. https://doi.org/10.1007/s00032-004-0032-7 doi: 10.1007/s00032-004-0032-7
    [32] S. I. Pohozaev, A. Tesei, Nonexistence of local solutions to semilinear partial differential inequalities, Ann. Inst. H. Poincaré Anal. Non Linéaire, 21 (2004), 487–502. https://doi.org/10.1016/j.anihpc.2003.06.002 doi: 10.1016/j.anihpc.2003.06.002
    [33] F. Punzo, A. Tesei, On a semilinear parabolic equation with inverse-square potential, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur., 21 (2010), 359–396. https://doi.org/10.4171/RLM/578 doi: 10.4171/RLM/578
    [34] F. Punzo, Blow-up of solutions to semilinear parabolic equations on Riemannian manifolds with negative sectional curvature, J. Math. Anal. Appl., 387 (2012), 815–827. https://doi.org/10.1016/j.jmaa.2011.09.043 doi: 10.1016/j.jmaa.2011.09.043
    [35] P. Quittner, The decay of global solutions of a semilinear heat equation, Discrete Contin. Dyn. Syst., 21 (2008), 307–318. https://doi.org/10.3934/dcds.2008.21.307 doi: 10.3934/dcds.2008.21.307
    [36] P. Souplet, Morrey spaces and classification of global solutions for a supercritical semilinear heat equation in \mathbb R^N, J. Funct. Anal., 272 (2017), 2005–2037. https://doi.org/10.1016/j.jfa.2016.09.002 doi: 10.1016/j.jfa.2016.09.002
    [37] J. L. Vázquez, The porous medium equation: mathematical theory, Oxford: Oxford University Press, 2007. https://doi.org/10.1093/acprof:oso/9780198569039.001.0001
    [38] L. Véron, Effets régularisants de semi-groupes non linéaires dans des espaces de Banach, Ann. Fac. Sci. Toulouse Math. (5), 1 (1979), 171–200.
    [39] Z. Wang, J. Yin, A note on semilinear heat equation in hyperbolic space, J. Differ. Equations, 256 (2014), 1151–1156. https://doi.org/10.1016/j.jde.2013.10.011 doi: 10.1016/j.jde.2013.10.011
    [40] Z. Wang, J. Yin, Asymptotic behaviour of the lifespan of solutions for a semilinear heat equation in hyperbolic space, Proc. Roy. Soc. Edinb. A, 146 (2016), 1091–1114. https://doi.org/10.1017/S0308210515000785 doi: 10.1017/S0308210515000785
    [41] F. B. Weissler, L^p-energy and blow-up for a semilinear heat equation, Proc. Sympos. Pure Math., 45 (1986), 545–551.
    [42] E. Yanagida, Behavior of global solutions of the Fujita equation, Sugaku Expositions, 26 (2013), 129–147.
    [43] Q. S. Zhang, Blow-up results for nonlinear parabolic equations on manifolds, Duke Math. J., 97 (1999), 515–539. https://doi.org/10.1215/S0012-7094-99-09719-3 doi: 10.1215/S0012-7094-99-09719-3
  • This article has been cited by:

    1. Stefano Biagi, Fabio Punzo, Eugenio Vecchi, Global solutions to semilinear parabolic equations driven by mixed local–nonlocal operators, 2024, 0024-6093, 10.1112/blms.13196
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2046) PDF downloads(148) Cited by(1)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog