Lightweight high-performance pose recognition network: HR-LiteNet

1.   Introduction

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

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

The problem is posed in the Lebesgue spaces

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

where $C_{s, p} > 0$ is a constant and $p^*: = \frac{pN}{N-p}$. In some cases we also assume that the Poincaré inequality is valid, that is

for some $C_p > 0$. 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 ${\mathbb{R}}^N$ (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 $M = {\mathbb{R}}^N$, 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 $\sigma < 1+\frac2N$, while global existence holds, for $\sigma > 1+ \frac 2N$, provided the initial datum is small enough in a suitable sense. Furthermore, the critical exponent $\sigma = 1+\frac 2N$, 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 $M = {\mathbb{R}}^N$ 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 $M = {\mathbb{R}}^N$ and

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

In ^[27] problem

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.

2.   Statements of the main results

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

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:

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

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

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

where

(ⅱ) Let $\sigma_0\le q < \infty$. If

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

where

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

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

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:

so that

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

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

Similarly, in (2.6), we can take

where

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

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

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 $q_0$:

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

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

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

where

Moreover, let $s\le q < \infty$ and

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

where

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

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

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:

so that

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

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

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

3.   $L^q-L^q$ and $L^{q_0}-L^{q}$ estimates

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 $k\in {\mathbb{R}}^+$

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

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

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:

3.1.   $L^q-L^q$ estimate for $\sigma > \sigma_0$

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

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

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 $u^{q-1}$,

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

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

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

Combining (3.6) and (3.7) we get

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

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

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

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

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

Thus we get

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.

3.2.   $L^{q_0}-L^q$ estimate for $\sigma > \sigma_0$

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

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

where

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

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

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

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

Observe that

We define

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

The latter can be rewritten as

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

We define

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

and

Consider the function

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

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

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

Let us set

Then (3.24) becomes

We define

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

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

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

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

where

Observe that

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 $u_0\equiv 0$.

3.3.   $L^q-L^q$ estimates for $\sigma > p-1$

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

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

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

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

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

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

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

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

where

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

Hence, we have

By substituting (3.38) into (3.36) we obtain

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

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

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

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

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

Thus we get

Iterating this procedure we obtain the thesis.

3.4.   $L^{q_0}-L^q$ estimate for $\sigma > p-1$

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

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

where

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

Observe that

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

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

where we have made use of inequality

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

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

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

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

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

Observe that

We define

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

The latter can be rewritten as

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

Consider the function

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

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

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

Let us set

Then (3.53) becomes

We define

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

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

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

where

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

4.   Auxiliary results

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

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

and the cylinders

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

Finally, we define

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

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

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:

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

Moreover,

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

Let us define

thus (4.10) reads

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

Thus (4.11) can be rewritten as

Let us now define

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

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:

By substituting (4.13) into (4.12) we obtain

To proceed further, observe that

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

The thesis follows, letting

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

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

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:

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

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:

Define

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

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

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

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

and that

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 $t\in(0, T)$,

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

for all $t\in(0, T)$.

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

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

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

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

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

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

Let us now set in (4.23):

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 $S(t)\le1$,

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

Furthermore, observe that

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

Let us finally define

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

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

provided

Now, (4.29), in turn, reads

Moreover, (4.30) is fulfilled since

This concludes the proof.

5.   Proof of Theorem 2.2

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

Then

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

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

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.

6.   Proof of Theorem 2.4

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

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

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

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

Define

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

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

Observe that

hence

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

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

Proof. Due to Lemma 6.1,

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

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

and

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

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).

7.   Proof of Theorem 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

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

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

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

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

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

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

Proof. We suppose by contradiction that

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

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

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

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

where we have set

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

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

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

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 $\Gamma = \Gamma(\sigma, p, N, q, s, C_{s, p}, C_p) > 0$

Proof. Due to Lemma 7.2,

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

Observing that

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

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).

8.   Porous medium equation with reaction

We now consider the following nonlinear reaction-diffusion problem:

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

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:

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:

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

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:

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

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

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

where

(ⅱ) Let $\sigma_1\le q < \infty$ and

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

where

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

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

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

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

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

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

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

where

Moreover, let $s\le q < \infty$ and

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

where

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

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

Acknowledgments

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).

Conflict of interest

The authors declare no conflict of interest.