Stability and optimization in structured population models on graphs

  • We prove existence and uniqueness of solutions, continuous dependence from the initial datum and stability with respect to the boundary condition in a class of initial--boundary value problems for systems of balance laws. The particular choice of the boundary condition allows to comprehend models with very different structures. In particular, we consider a juvenile-adult model, the problem of the optimal mating ratio and a model for the optimal management of biological resources. The stability result obtained allows to tackle various optimal management/control problems, providing sufficient conditions for the existence of optimal choices/controls.

    Citation: Rinaldo M. Colombo, Mauro Garavello. Stability and optimization in structured population models on graphs[J]. Mathematical Biosciences and Engineering, 2015, 12(2): 311-335. doi: 10.3934/mbe.2015.12.311

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  • We prove existence and uniqueness of solutions, continuous dependence from the initial datum and stability with respect to the boundary condition in a class of initial--boundary value problems for systems of balance laws. The particular choice of the boundary condition allows to comprehend models with very different structures. In particular, we consider a juvenile-adult model, the problem of the optimal mating ratio and a model for the optimal management of biological resources. The stability result obtained allows to tackle various optimal management/control problems, providing sufficient conditions for the existence of optimal choices/controls.


    We consider the Cauchy problem for the nonlinear damped wave equation with linear damping

    $ uttΔu+ut=|u|p+|u|q+w(x),t>0,xRN,
    $
    (1.1)

    where $ N\geq 1 $, $ p, q > 1 $, $ w\in L^1_{loc}(\mathbb{R}^N) $, $ w\geq 0 $ and $ w\not\equiv 0 $. Namely, we are concerned with the existence and nonexistence of global weak solutions to (1.1). We mention below some motivations for studying problems of type (1.1).

    Consider the semilinear damped wave equation

    $ uttΔu+ut=|u|p,t>0,xRN
    $
    (1.2)

    under the initial conditions

    $ (u(0,x),ut(0,x))=(u0(x),u1(x)),xRN.
    $
    (1.3)

    In [12], a Fujita-type result was obtained for the problem (1.2) and (1.3). Namely, it was shown that,

    (ⅰ) if $ 1 < p < 1+\frac{2}{N} $ and $ \int_{\mathbb{R}^N}u_j(x)\, dx > 0 $, $ j = 0, 1 $, then the problem (1.2) and (1.3) admits no global solution;

    (ⅱ) if $ 1+\frac{2}{N} < p < N $ for $ N\geq 3 $, and $ 1+\frac{2}{N} < p < \infty $ for $ N\in\{1, 2\} $, then the problem (1.2) and (1.3) admits a unique global solution for suitable initial values.

    The proof of part (ⅰ) makes use of the fundamental solution of the operator $ (\partial_{tt} - \Delta + \partial_t)^k $. In [15], it was shown that the exponent $ 1+\frac{2}{N} $ belongs to the blow-up case (ⅰ). Notice that $ 1+\frac{2}{N} $ is also the Fujita critical exponent for the semilinear heat equation $ u_t-\Delta u = |u|^p $ (see [3]). In [6], the authors considered the problem

    $ utt+(1)m|x|αΔmu+ut=f(t,x)|u|p+w(t,x),t>0,xRN
    $
    (1.4)

    under the initial conditions (1.3), where $ m $ is a positive natural number, $ \alpha\geq 0 $, and $ f(t, x)\geq 0 $ is a given function satisfying a certain condition. Using the test function method (see e.g. [11]), sufficient conditions for the nonexistence of a global weak solution to the problem (1.4) and (1.3) are obtained. Notice that in [6], the influence of the inhomogeneous term $ w(t, x) $ on the critical behavior of the problem (1.4) and (1.3) was not investigated. Recently, in [5], the authors investigated the inhomogeneous problem

    $ uttΔu+ut=|u|p+w(x),t>0,xRN,
    $
    (1.5)

    where $ w\in L^1_{loc}(\mathbb{R}^N) $ and $ w\not\equiv 0 $. It was shown that in the case $ N\geq 3 $, the critical exponent for the problem (1.5) jumps from $ 1+\frac{2}{N} $ (the critical exponent for the problem (1.2)) to the bigger exponent $ 1+\frac{2}{N-2} $. Notice that a similar phenomenon was observed for the inhomogeneous semilnear heat equation $ u_t-\Delta u = |u|^p+w(x) $ (see [14]). For other results related to the blow-up of solutions to damped wave equations, see, for example [1,2,4,7,8,9,13] and the references therein.

    In this paper, motivated by the above mentioned works, our aim is to study the effect of the gradient term $ |\nabla u|^q $ on the critical behavior of problem (1.5). We first obtain the Fujita critical exponent for the problem (1.1). Next, we determine its second critical exponent in the sense of Lee and Ni [10].

    Before stating the main results, let us provide the notion of solutions to problem (1.1). Let $ Q = (0, \infty)\times \mathbb{R}^N $. We denote by $ C_c^2(Q) $ the space of $ C^2 $ real valued functions compactly supported in $ Q $.

    Definition 1.1. We say that $ u = u(t, x) $ is a global weak solution to (1.1), if

    $ (u,\nabla u)\in L^p_{loc}(Q)\times L^q_{loc}(Q) $

    and

    $ Q(|u|p+|u|q)φdxdt+Qw(x)φdxdt=QuφttdxdtQuΔφdxdtQuφtdxdt,
    $
    (1.6)

    for all $ \varphi \in C_c^2(Q) $.

    The first main result is the following Fujita-type theorem.

    Theorem 1.1. Let $ N\geq 1 $, $ p, q > 1 $, $ w\in L^1_{loc}(\mathbb{R}^N) $, $ w\geq 0 $ and $ w\not\equiv 0 $.

    (i) If $ N\in\{1, 2\} $, for all $ p, q > 1 $, problem (1.1) admits no global weak solution.

    (ii) Let $ N\geq 3 $. If

    $ 1 \lt p \lt 1+\frac{2}{N-2}\quad\mathit{\mbox{or}}\quad 1 \lt q \lt 1+\frac{1}{N-1}, $

    then problem (1.1) admits no global weak solution.

    (iii) Let $ N\geq 3 $. If

    $ p \gt 1+\frac{2}{N-2}\quad\mathit{\mbox{and}}\quad q \gt 1+\frac{1}{N-1}, $

    then problem (1.1) admits global solutions for some $ w > 0 $.

    We mention below some remarks related to Theorem 1.1. For $ N\geq 3 $, let

    $ p_c(N,q) = \left\{1+2N2ifq>1+1N1,ifq<1+1N1.
    \right. $

    From Theorem 1.1, if $ 1 < p < p_c(N, q) $, then problem (1.1) admits no global weak solution, while if $ p > p_c(N, q) $, global weak solutions exist for some $ w > 0 $. This shows that $ p_c(N, q) $ is the Fujita critical exponent for the problem (1.1). Notice that in the case $ N\geq 3 $ and $ q > 1+\frac{1}{N-1} $, $ p_c(N, q) $ is also the critical exponent for $ u_{tt}-\Delta u = |u|^p+w(x) $ (see [5]). This shows that in the range $ q > 1+\frac{1}{N-1} $, the gradient term $ |\nabla u|^q $ has no influence on the critical behavior of $ u_{tt}-\Delta u = |u|^p+w(x) $.

    It is interesting to note that the gradient term $ |\nabla u|^q $ induces an interesting phenomenon of discontinuity of the Fujita critical exponent $ p_c(N, q) $ jumping from $ 1+\frac{2}{N-2} $ to $ \infty $ as $ q $ reaches the value $ 1+\frac {1}{N-1} $ from above.

    Next, for $ N\geq 3 $ and $ \sigma < N $, we introduce the sets

    $ I_\sigma^+ = \left\{w\in C(\mathbb{R}^N):\, w\geq 0,\, |x|^{-\sigma} = O(w(x)) \mbox{ as }|x|\to \infty\right\} $

    and

    $ I_\sigma^- = \left\{w\in C(\mathbb{R}^N):\, w \gt 0,\, w(x) = O(|x|^{-\sigma})\mbox{ as }|x|\to \infty\right\}. $

    The next result provides the second critical exponent for the problem (1.1) in the sense of Lee and Ni [10].

    Theorem 1.2. Let $ N\geq 3 $, $ p > 1+\frac{2}{N-2} $ and $ q > 1+\frac{1}{N-1} $.

    (i) If

    $ \sigma \lt \max\left\{\frac{2p}{p-1}, \frac{q}{q-1}\right\} $

    and $ w\in I_\sigma^+ $, then problem (1.1) admits no global weak solution.

    (ii) If

    $ \sigma \geq \max\left\{\frac{2p}{p-1}, \frac{q}{q-1}\right\} , $

    then problem (1.1) admits global solutions for some $ w\in I_\sigma^- $.

    From Theorem 1.2, if $ N\geq 3 $, $ p > 1+\frac{2}{N-2} $ and $ q > 1+\frac{1}{N-1} $, then problem (1.1) admits a second critical exponent, namely

    $ \sigma^* = \max\left\{\frac{2p}{p-1}, \frac{q}{q-1}\right\}. $

    The rest of the paper is organized as follows. In Section 2, some preliminary estimates are provided. Section 3 is devoted to the study of the Fujita critical exponent for the problem (1.1). Namely, Theorem 1.1 is proved. In Section 4, we study the second critical exponent for the problem (1.1) in the sense of Lee and Ni. Namely, we prove Theorem 1.2.

    Consider two cut-off functions $ f, g \in C^\infty([0, \infty)) $ satisfying

    $ f\geq 0,\quad f\not\equiv 0,\quad \mbox{supp}(f) \subset (0,1) $

    and

    $ 0\leq g\leq 1,\quad g(s) = \left\{1if0s1,0ifs2.
    \right. $

    For $ T > 0 $, we introduce the function

    $ ξT(t,x)=f(tT)g(|x|2T2ρ)=F(t)G(x),(t,x)Q,
    $
    (2.1)

    where $ \ell\geq 2 $ and $ \rho > 0 $ are constants to be chosen. It can be easily seen that

    $ \xi_T\geq 0\quad\mbox{and}\quad \xi_T\in C_c^2(Q). $

    Throughout this paper, the letter $ C $ denotes various positive constants depending only on known quantities.

    Lemma 2.1. Let $ \kappa > 1 $ and $ \ell > \frac{2\kappa}{\kappa-1} $. Then

    $ Q|ξT|1κ1|ΔξT|κκ1dxdtCT2ρκκ1+Nρ+1.
    $
    (2.2)

    Proof. By (2.1), one has

    $ Q|ξT|1κ1|ΔξT|κκ1dxdt=(0F(t)dt)(RN|G|1κ1|ΔG|κκ1dx).
    $
    (2.3)

    Using the properties of the cut-off function $ f $, one obtains

    $ 0F(t)dt=0f(tT)dt=T0f(tT)dt=T10f(s)ds,
    $

    which shows that

    $ 0F(t)dt=CT.
    $
    (2.4)

    Next, using the properties of the cut-off function $ g $, one obtains

    $ RN|G|1κ1|ΔG|κκ1dx=Tρ<|x|<2Tρ|G|1κ1|ΔG|κκ1dx
    $
    (2.5)

    On the other hand, for $ T^\rho < |x| < \sqrt 2T^\rho $, an elementary calculation yields

    $ ΔG(x)=Δ[g(r2T2ρ)],r=|x|=(d2dr2+N1rddr)g(r2T2ρ)=2T2ρg(r2T2ρ)2θ(x),
    $

    where

    $ \theta(x)\\ = N g\left(\frac{r^2}{T^{2\rho}}\right)g'\left(\frac{r^2}{T^{2\rho}}\right)+2(\ell-1)T^{-2\rho}r^2g\left(\frac{r^2}{T^{2\rho}}\right)g'\left(\frac{r^2}{T^{2\rho}}\right)^2+2T^{-2\rho}r^2g\left(\frac{r^2}{T^{2\rho}}\right)g''\left(\frac{r^2}{T^{2\rho}}\right). $

    Notice that since $ T^\rho < r < \sqrt 2T^\rho $, one deduces that

    $ |\theta(x)|\leq C. $

    Hence, it holds that

    $ |\Delta G(x)| \leq C T^{-2\rho}g\left(\frac{|x|^2}{T^{2\rho}}\right)^{\ell-2},\quad T^\rho \lt |x| \lt \sqrt 2T^\rho $

    and

    $ |G(x)|^{\frac{-1}{\kappa-1}}|\Delta G(x)|^{\frac{\kappa}{\kappa-1}}\leq C T^{\frac{-2\rho \kappa}{\kappa-1}} g\left(\frac{|x|^2}{T^{2\rho}}\right)^{\ell-\frac{2\kappa}{\kappa-1}},\quad T^\rho \lt |x| \lt \sqrt 2T^\rho. $

    Therefore, by (2.5) and using the change of variable $ x = T^\rho y $, one obtains

    $ RN|G|1κ1|ΔG|κκ1dxCT2ρκκ1Tρ<|x|<2Tρg(|x|2T2ρ)2κκ1dx=CT2ρκκ1+Nρ1<|y|<2g(|y|2)2κκ1dy,
    $

    which yields (notice that $ \ell > \frac{2\kappa}{\kappa-1} $)

    $ RN|G|1κ1|ΔG|κκ1dxCT2ρκκ1+Nρ.
    $
    (2.6)

    Finally, using (2.3), (2.4) and (2.6), (2.2) follows.

    Lemma 2.2. Let $ \kappa > 1 $ and $ \ell > \frac{\kappa}{\kappa-1} $. Then

    $ Q|ξT|1κ1|ξT|κκ1dxdtCTρκκ1+Nρ+1.
    $
    (2.7)

    Proof. By (2.1) and the properties of the cut-off function $ g $, one has

    $ Q|ξT|1κ1|ξT|κκ1dxdt=(0F(t)dt)(Tρ<|x|<2Tρ|G|1κ1|G|κκ1dx).
    $
    (2.8)

    On the other hand, for $ T^\rho < |x| < \sqrt 2 T^\rho $, an elementary calculation shows that

    $ |G(x)|=2T2ρ|x||g(|x|2T2ρ)|g(|x|2T2ρ)122Tρ|g(|x|2T2ρ)|g(|x|2T2ρ)1CTρg(|x|2T2ρ)1,
    $

    which yields

    $ |G|^{\frac{-1}{\kappa-1}}|\nabla G|^{\frac{\kappa}{\kappa-1}}\leq C T^{\frac{-\rho \kappa}{\kappa-1}} g\left(\frac{|x|^2}{T^{2\rho}}\right)^{\ell-\frac{\kappa}{\kappa-1}},\quad T^\rho \lt |x| \lt \sqrt 2T^\rho. $

    Therefore, using the change of variable $ x = T^\rho y $ and the fact that $ \ell > \frac{\kappa}{\kappa-1} $, one obtains

    $ Tρ<|x|<2Tρ|G|1κ1|G|κκ1dxCTρκκ1Tρ<|x|<2Tρg(|x|2T2ρ)κκ1dx=CTρκκ1+Nρ1<|y|<2g(|y|2)κκ1dy,
    $

    i.e.

    $ Tρ<|x|<2Tρ|G|1κ1|G|κκ1dxCTρκκ1+Nρ.
    $
    (2.9)

    Using (2.4), (2.8) and (2.9), (2.7) follows.

    Lemma 2.3. Let $ \kappa > 1 $ and $ \ell > \frac{2\kappa}{\kappa-1} $. Then

    $ Q|ξT|1κ1|ξTtt|κκ1dxdtCT2κκ1+1+Nρ.
    $
    (2.10)

    Proof. By (2.1) and the properties of the cut-off functions $ f $ and $ g $, one has

    $ Q|ξT|1κ1|ξTtt|κκ1dxdt=(T0F(t)1κ1|F(t)|κκ1dt)(0<|x|<2TρG(x)dx).
    $
    (2.11)

    An elementary calculation shows that

    $ F(t)^{\frac{-1}{\kappa-1}}|F''(t)|^{\frac{\kappa}{\kappa-1}}\leq C T^{\frac{-2\kappa}{\kappa-1}} f\left(\frac{t}{T}\right)^{\ell-\frac{2\kappa}{\kappa-1}},\quad 0 \lt t \lt T, $

    which yields (since $ \ell > \frac{2\kappa}{\kappa-1} $)

    $ T0F(t)1κ1|F(t)|κκ1dtCT2κκ1+1.
    $
    (2.12)

    On the other hand, using the change of variable $ x = T^\rho y $, one obtains

    $ 0<|x|<2TρG(x)dx=0<|x|<2Tρg(|x|2T2ρ)dx=TNρ0<|y|<2g(|y|2)dy,
    $

    which yields

    $ 0<|x|<2TρG(x)dxCTNρ.
    $
    (2.13)

    Therefore, using (2.11), (2.12) and (2.13), (2.10) follows.

    Lemma 2.4. Let $ \kappa > 1 $ and $ \ell > \frac{\kappa}{\kappa-1} $. Then

    $ Q|ξT|1κ1|ξTt|κκ1dxdtCTκκ1+1+Nρ.
    $
    (2.14)

    Proof. By (2.1) and the properties of the cut-off functions $ f $ and $ g $, one has

    $ Q|ξT|1κ1|ξTt|κκ1dxdt=(T0F(t)1κ1|F(t)|κκ1dt)(0<|x|<2TρG(x)dx).
    $
    (2.15)

    An elementary calculation shows that

    $ F(t)^{\frac{-1}{\kappa-1}}|F'(t)|^{\frac{\kappa}{\kappa-1}}\leq C T^{\frac{-\kappa}{\kappa-1}} f\left(\frac{t}{T}\right)^{\ell-\frac{\kappa}{\kappa-1}},\quad 0 \lt t \lt T, $

    which yields (since $ \ell > \frac{\kappa}{\kappa-1} $)

    $ T0F(t)1κ1|F(t)|κκ1dtCT1κκ1.
    $
    (2.16)

    Therefore, using (2.13), (2.15) and (2.16), (2.14) follows.

    Proposition 3.1. Suppose that problem (1.1) admits a global weak solution $ u $ with $ (u, \nabla u)\in L^p_{loc}(Q)\times L^q_{loc}(Q) $. Then there exists a constant $ C > 0 $ such that

    $ Qw(x)φdxdtCmin{A(φ),B(φ)},
    $
    (3.1)

    for all $ \varphi\in C_c^2(Q) $, $ \varphi\geq 0 $, where

    $ A(\varphi) = \int_Q \varphi^{\frac{-1}{p-1}}|\varphi_{tt}|^{\frac{p}{p-1}}\,dx\,dt+\int_Q \varphi^{\frac{-1}{p-1}}|\varphi_{t}|^{\frac{p}{p-1}}\,dx\,dt +\int_Q \varphi^{\frac{-1}{p-1}}|\Delta \varphi|^{\frac{p}{p-1}}\,dx\,dt $

    and

    $ B(\varphi) = \int_Q \varphi^{\frac{-1}{p-1}}|\varphi_{tt}|^{\frac{p}{p-1}}\,dx\,dt+\int_Q \varphi^{\frac{-1}{p-1}}|\varphi_{t}|^{\frac{p}{p-1}}\,dx\,dt +\int_Q \varphi^{\frac{-1}{q-1}}|\nabla \varphi|^{\frac{q}{q-1}}\,dx\,dt. $

    Proof. Let $ u $ be a global weak solution to problem (1.1) with

    $ (u,\nabla u)\in L^p_{loc}(Q)\times L^q_{loc}(Q). $

    Let $ \varphi\in C_c^2(Q) $, $ \varphi\geq 0 $. Using (1.6), one obtains

    $ Q|u|pφdxdt+Qw(x)φdxdtQ|u||φtt|dxdt+Q|u||Δφ|dxdt+Q|u||φt|dxdt.
    $
    (3.2)

    On the other hand, by $ \varepsilon $-Young inequality, $ 0 < \varepsilon < \frac{1}{3} $, one has

    $ Q|u||φtt|dxdt=Q(|u|φ1p)(φ1p|φtt|)dxdtεQ|u|pφdxdt+CQφ1p1|φtt|pp1dxdt.
    $
    (3.3)

    Here and below, $ C $ denotes a positive generic constant, whose value may change from line to line. Similarly, one has

    $ Q|u||Δφ|dxdtεQ|u|pφdxdt+CQφ1p1|Δφ|pp1dxdt
    $
    (3.4)

    and

    $ Q|u||φt|dxdtεQ|u|pφdxdt+CQφ1p1|φt|pp1dxdt
    $
    (3.5)

    Therefore, using (3.2), (3.3), (3.4) and (3.5), it holds that

    $ (1-3\varepsilon)\int_Q |u|^p\varphi\,dx\,dt+ \int_Q w(x)\varphi\,dx\,dt\leq C A(\varphi). $

    Since $ 1-3\varepsilon > 0 $, one deduces that

    $ Qw(x)φdxdtCA(φ).
    $
    (3.6)

    Again, by (1.6), one has

    $ Q(|u|p+|u|q)φdxdt+Qw(x)φdxdt=QuφttdxdtQuΔφdxdtQuφtdxdt=Quφttdxdt+QuφdxdtQuφtdxdt,
    $

    where $ \cdot $ denotes the inner product in $ \mathbb{R}^N $. Hence, it holds that

    $ Q(|u|p+|u|q)φdxdt+Qw(x)φdxdtQ|u||φtt|dxdt+Q|u||φ|dxdt+Q|u||φt|dxdt.
    $
    (3.7)

    By $ \varepsilon $-Young inequality ($ \varepsilon = \frac{1}{2} $), one obtains

    $ Q|u||φtt|dxdt12Q|u|pφdxdt+CQφ1p1|φtt|pp1dxdt,
    $
    (3.8)
    $ Q|u||φt|dxdt12Q|u|pφdxdt+CQφ1p1|φt|pp1dxdt,
    $
    (3.9)
    $ Q|u||φ|dxdt12Q|u|qφdxdt+CQφ1q1|φ|qq1dxdt.
    $
    (3.10)

    Next, using (3.7), (3.8), (3.9) and (3.10), one deduces that

    $ \frac{1}{2} \int_Q |\nabla u|^q\varphi\,dx\,dt+\int_{Q} w(x)\varphi \,\,dx\,dt\leq CB(\varphi), $

    which yields

    $ Qw(x)φdxdtCB(φ).
    $
    (3.11)

    Finally, combining (3.6) with (3.11), (3.1) follows.

    Proposition 3.2. Suppose that problem (1.1) admits a global weak solution $ u $ with $ (u, \nabla u)\in L^p_{loc}(Q)\times L^q_{loc}(Q) $. Then

    $ Qw(x)ξTdxdtCmin{A(ξT),B(ξT)},
    $
    (3.12)

    for all $ T > 0 $, where $ \xi_T $ is defined by (2.1).

    Proof. Since for all $ T > 0 $, $ \xi_T\in C_c^2(Q) $, $ \xi_T\geq 0 $, taking $ \varphi = \xi_T $ in (3.1), (3.12) follows.

    Proposition 3.3. Let $ w\in L^1_{loc}(\mathbb{R}^N) $, $ w\geq 0 $ and $ w\not\equiv 0 $. Then, for sufficiently large $ T $,

    $ Qw(x)ξTdxdtCT,
    $
    (3.13)

    where $ \xi_T $ is defined by (2.1).

    Proof. By (2.1) and the properties of the cut-off functions $ f $ and $ g $, one has

    $ Qw(x)ξTdxdt=(T0f(tT)dt)(0<|x|<2Tρw(x)g(|x|2T2ρ)dx).
    $
    (3.14)

    On the other hand,

    $ T0f(tT)dt=T10f(s)ds.
    $
    (3.15)

    Moreover, for sufficiently large $ T $ (since $ w, g\geq 0 $),

    $ 0<|x|<2Tρw(x)g(|x|2T2ρ)dx0<|x|<1w(x)g(|x|2T2ρ)dx.
    $
    (3.16)

    Notice that since $ w\in L^1_{loc}(\mathbb{R}^N) $, $ w\geq 0 $, $ w\not\equiv 0 $, and using the properties of the cut-off function $ g $, by the domianted convergence theorem, one has

    $ \lim\limits_{T\to \infty}\int_{0 \lt |x| \lt 1} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx = \int_{0 \lt |x| \lt 1} w(x)\,dx \gt 0. $

    Hence, for sufficiently large $ T $,

    $ 0<|x|<1w(x)g(|x|2T2ρ)dxC.
    $
    (3.17)

    Using (3.14), (3.15), (3.16) and (3.17), (3.13) follows.

    Now, we are ready to prove the Fujita-type result given by Theorem 1.1. The proof is by contradiction and makes use of the nonlinear capacity method, which was developed by Mitidieri and Pohozaev (see e.g. [11]).

    Proof of Theorem 1.1. (ⅰ) Suppose that problem (1.1) admits a global weak solution $ u $ with $ (u, \nabla u)\in L^p_{loc}(Q)\times L^q_{loc}(Q) $. By Propositions 3.2 and 3.3, for sufficiently large $ T $, one has

    $ CTmin{A(ξT),B(ξT)},
    $
    (3.18)

    where $ \xi_T $ is defined by (2.1),

    $ A(ξT)=Qξ1p1T|ξTtt|pp1dxdt+Qξ1p1T|ξTt|pp1dxdt+Qξ1p1T|ΔξT|pp1dxdt
    $
    (3.19)

    and

    $ B(ξT)=Qφ1p1|ξTtt|pp1dxdt+Qξ1p1T|ξTt|pp1dxdt+Qξ1q1T|ξT|qq1dxdt.
    $
    (3.20)

    Taking $ \ell > \max\left\{\frac{2p}{p-1}, \frac{q}{q-1}\right\} $, using Lemmas 2.1, 2.3, 2.4 with $ \kappa = p $, and Lemma 2.2 with $ \kappa = q $, one obtains the following estimates

    $ Qξ1p1T|ξTtt|pp1dxdtCT2pp1+1+Nρ,
    $
    (3.21)
    $ Qξ1p1T|ξTt|pp1dxdtCTpp1+1+Nρ,
    $
    (3.22)
    $ Qξ1p1T|ΔξT|pp1dxdtCT2ρpp1+Nρ+1,
    $
    (3.23)
    $ Qξ1q1T|ξT|qq1dxdtCTρqq1+Nρ+1.
    $
    (3.24)

    Hence, by (3.19), (3.21), (3.22) and (3.23), one deduces that

    $ A(\xi_T)\leq C\left(T^{\frac{-2p}{p-1}+1+N\rho}+ T^{\frac{-p}{p-1}+1+N\rho}+T^{\frac{-2\rho p}{p-1}+N\rho+1}\right). $

    Observe that

    $ 2pp1+1+Nρ<pp1+1+Nρ.
    $
    (3.25)

    So, for sufficiently large $ T $, one deduces that

    $ A(\xi_T)\leq C\left(T^{\frac{-p}{p-1}+1+N\rho}+T^{\frac{-2\rho p}{p-1}+N\rho+1}\right). $

    Notice that the above estimate holds for all $ \rho > 0 $. In particular, when $ \rho = \frac{1}{2} $, it holds that

    $ A(ξT)CTpp1+1+N2
    $
    (3.26)

    Next, by (3.20), (3.21), (3.22), (3.24) and (3.25), one deduces that

    $ B(\xi_T)\leq C\left(T^{\frac{-p}{p-1}+1+N\rho}+T^{\frac{-\rho q}{q-1}+N\rho+1}\right). $

    Similarly, the above inequality holds for all $ \rho > 0 $. In particular, when $ \rho = \frac{p(q-1)}{q(p-1)} $, it holds that

    $ B(ξT)CTpp1+1+Np(q1)q(p1).
    $
    (3.27)

    Therefore, it follows from (3.18), (3.26) and (3.27) that

    $ 0 \lt C\leq \min \left\{T^{\frac{-p}{p-1}+\frac{N}{2}}, T^{\frac{-p}{p-1}+\frac{Np(q-1)}{q(p-1)}}\right\}, $

    which yields

    $ 0<CTpp1+N2:=Tα(N)
    $
    (3.28)

    and

    $ 0<CTpp1+Np(q1)q(p1):=Tβ(N).
    $
    (3.29)

    Notice that for $ N\in\{1, 2\} $, one has

    $ \alpha(N) = \left\{p12(p1)ifN=1,1p1ifN=2,
    \right. $

    which shows that

    $ \alpha(N) \lt 0,\quad N\in \{1,2\}. $

    Hence, for $ N\in\{1, 2\} $, passing to the limit as $ T\to \infty $ in (3.28), a contradiction follows ($ 0 < C\leq 0 $). This proves part (ⅰ) of Theorem 1.1.

    (ⅱ) Let $ N\geq 3 $. In this case, one has

    $ \alpha(N) \lt 0 \Longleftrightarrow p \lt 1+\frac{2}{N-2}. $

    Hence, if $ p < 1+\frac{2}{N-2} $, passing to the limit as $ T\to \infty $ in (3.28), a contradiction follows. Furthermore, one has

    $ \beta(N) \lt 0 \Longleftrightarrow q \lt 1+\frac{1}{N-1}. $

    Hence, if $ q < 1+\frac{1}{N-1} $, passing to the limit as $ T\to \infty $ in (3.29), a contradiction follows. Therefore, we proved part (ⅱ) of Theorem 1.1.

    (ⅲ) Let

    $ p>1+2N2andq>1+1N1.
    $
    (3.30)

    Let

    $ u(x)=ε(1+r2)δ,r=|x|,xRN,
    $
    (3.31)

    where

    $ max{1p1,2q2(q1)}<δ<N22
    $
    (3.32)

    and

    $ 0<ε<min{1,(2δ(N2δ2)1+2qδq)1min{p,q}1}.
    $
    (3.33)

    Note that due to (3.30), the set of $ \delta $ satisfying (3.32) is nonempty. Let

    $ w(x) = -\Delta u-|u|^p-|\nabla u|^q,\quad x\in \mathbb{R}^N. $

    Elementary calculations yield

    $ w(x)=2δε[(1+r2)δ12r2(δ+1)(1+r2)δ2+(N1)(1+r2)δ1]εp(1+r2)δp2qδqεqrq(1+r2)(δ+1)q.
    $
    (3.34)

    Hence, one obtains

    $ w(x)2δε[(1+r2)δ12(δ+1)(1+r2)δ1+(N1)(1+r2)δ1]εp(1+r2)δp2qδqεq(1+r2)(δ+1)q+q2=2δε(N2δ2)(1+r2)δ1εp(1+r2)δp2qεqδq(1+r2)(δ+12)q.
    $

    Next, using (3.32) and (3.33), one deduces that

    $ w(x)\geq \varepsilon \left[2\delta (N-2\delta-2)-\varepsilon^{p-1}-2^q\varepsilon^{q-1}\delta^q\right](1+r^2)^{-\delta-1} \gt 0. $

    Therefore, for any $ \delta $ and $ \varepsilon $ satisfying respectively (3.32) and (3.33), the function $ u $ defined by (3.31) is a stationary solution (then global solution) to (1.1) for some $ w > 0 $. This proves part (ⅲ) of Theorem 1.1.

    Proposition 4.1. Let $ N\geq 3 $, $ \sigma < N $ and $ w\in I_\sigma^+ $. Then, for sufficiently large $ T $,

    $ Qw(x)ξTdxdtCTρ(Nσ)+1.
    $
    (4.1)

    where $ \xi_T $ is defined by (2.1).

    Proof. By (3.14) and (3.15), one has

    $ Qw(x)ξTdxdt=CT0<|x|<2Tρw(x)g(|x|2T2ρ)dx.
    $
    (4.2)

    On the other hand, by definition of $ I_\sigma^+ $, and using that $ g(s) = 1 $, $ 0\leq s\leq 1 $, for sufficiently large $ T $, one obtains (since $ w, g\geq 0 $)

    $ 0<|x|<2Tρw(x)g(|x|2T2ρ)dx0<|x|<Tρw(x)g(|x|2T2ρ)dx=0<|x|<Tρw(x)dxTρ2<|x|<Tρw(x)dxCTρ2<|x|<Tρ|x|σdx=CTρ(Nσ).
    $

    Hence, using (4.2), (4.1) follows.

    Now, we are ready to prove the new critical behavior for the problem (1.1) stated by Theorem 1.2.

    Proof of Theorem 1.2. (ⅰ) Suppose that problem (1.1) admits a global weak solution $ u $ with $ (u, \nabla u)\in L^p_{loc}(Q)\times L^q_{loc}(Q) $. By Propositions 3.2 and 4.1, for sufficiently large $ T $, one obtains

    $ CTρ(Nσ)+1min{A(ξT),B(ξT)},
    $
    (4.3)

    where $ \xi_T $, $ A(\xi_T) $ and $ B(\xi_T) $ are defined respectively by (2.1), (3.19) and (3.20). Next, using (3.26) and (4.3) with $ \rho = \frac{1}{2} $, one deduces that

    $ 0<CTpp1+σ2.
    $
    (4.4)

    Observe that

    $ \frac{-p}{p-1}+\frac{\sigma}{2} \lt 0 \Longleftrightarrow \sigma \lt \frac{2p}{p-1}. $

    Hence, if $ \sigma < \frac{2p}{p-1} $, passing to the limit as $ T\to \infty $ in (4.4), a contradiction follows. Furthermore, using (3.27) and (4.3) with $ \rho = \frac{p(q-1)}{q(p-1)} $, one deduces that

    $ 0<CTpp1(σ(q1)q1).
    $
    (4.5)

    Observe that

    $ \frac{p}{p-1}\left(\frac{\sigma(q-1)}{q}-1\right) \lt 0\Longleftrightarrow \sigma \lt \frac{q}{q-1}. $

    Hence, if $ \sigma < \frac{q}{q-1} $, passing to the limit as $ T\to \infty $ in (4.5), a contradiction follows. Therefore, part (ⅰ) of Theorem 1.2 is proved.

    (ⅱ) Let

    $ σmax{2pp1,qq1}.
    $
    (4.6)

    Let $ u $ be the function defined by (3.31), where

    $ σ22<δ<N22
    $
    (4.7)

    and $ \varepsilon $ satisfies (3.33). Notice that since $ \sigma < N $, the set of $ \delta $ satisfying (4.7) is nonempty. Moreover, due to (4.6) and (4.7), $ \delta $ satisfies also (3.32). Hence, from the proof of part (iii) of Theorem 1.2, one deduces that

    $ w(x) = -\Delta u-|u|^p-|\nabla u|^q \gt 0,\quad x\in \mathbb{R}^N. $

    On the other hand, using (3.34) and (4.7), for $ |x| $ large enough, one obtains

    $ w(x)\leq C (1+|x|^2)^{-\delta-1} \leq C |x|^{-2\delta-2}\leq C|x|^{-\delta}, $

    which proves that $ w\in I_\sigma^- $. This proves part (ⅱ) of Theorem 1.2.

    We investigated the large-time behavior of solutions to the nonlinear damped wave equation (1.1). In the case when $ N\in\{1, 2\} $, we proved that for all $ p > 1 $, problem (1.1) admits no global weak solution (in the sense of Definition 1). Notice that from [5], the same result holds for the problem without gradient term, namely problem (1.5). This shows that in the case $ N\in\{1, 2\} $, the nonlinearity $ |\nabla u|^q $ has no influence on the critical behavior of problem (1.5). In the case when $ N\geq 3 $, we proved that, if $ 1 < p < 1+\frac{2}{N-2} $ or $ 1 < q < 1+\frac{1}{N-1} $, then problem (1.1) admits no global weak solution, while if $ p > 1+\frac{2}{N-2} $ and $ q > 1+\frac{1}{N-1} $, global solutions exist for some $ w > 0 $. This shows that in this case, the Fujita critical exponent for the problem (1.1) is given by

    $ p_c(N,q) = \left\{1+2N2ifq>1+1N1,ifq<1+1N1.
    \right. $

    From this result, one observes two facts. First, in the range $ q > 1+\frac{1}{N-1} $, from [5], the critical exponent $ p_c(N, q) $ is also equal to the critical exponent for the problem without gradient term, which means that in this range of $ q $, the nonlinearity $ |\nabla u|^q $ has no influence on the critical behavior of problem (1.5). Secondly, one observes that the gradient term induces an interesting phenomenon of discontinuity of the Fujita critical exponent $ p_c(N, q) $ jumping from $ 1+\frac{2}{N-2} $ to $ \infty $ as $ q $ reaches the value $ 1+\frac {1}{N-1} $ from above. In the same case $ N\geq 3 $, we determined also the second critical exponent for the problem (1.1) in the sense of Lee and Ni [10], when $ p > 1+\frac{2}{N-2} $ and $ q > 1+\frac{1}{N-1} $. Namely, we proved that in this case, if $ \sigma < \max\left\{\frac{2p}{p-1}, \frac{q}{q-1}\right\} $ and $ w\in I_\sigma^+ $, then there is no global weak solution, while if $ \max\left\{\frac{2p}{p-1}, \frac{q}{q-1}\right\}\leq \sigma < N $, global solutions exist for some $ w\in I_\sigma^- $. This shows that the second critical exponent for the problem (1.1) in the sense of Lee and Ni is given by

    $ \sigma^* = \max\left\{\frac{2p}{p-1},\frac{q}{q-1}\right\}. $

    We end this section with the following open questions:

    (Q1). Find the first and second critical exponents for the system of damped wave equations with mixed nonlinearities

    $ {uttΔu+ut=|v|p1+|v|q1+w1(x)in(0,)×RN,vttΔv+vt=|u|p2+|u|q2+w2(x)in(0,)×RN,
    $

    where $ p_i, q_i > 1 $, $ w_i\in L^1_{loc}(\mathbb{R}^N) $, $ w_i\geq 0 $ and $ w_i\not\equiv 0 $, $ i = 1, 2 $.

    (Q2). Find the Fujita critical exponent for the problem (1.1) with $ w\equiv 0 $.

    (Q3) Find the Fujita critical exponent for the problem

    $ u_{tt}-\Delta u +u_t = \frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}|u(s,x)|^{p}\,ds +|\nabla u|^{q} +w(x),\quad t \gt 0,\,x\in \mathbb{R}^N, $

    where $ N\geq 1 $, $ p, q > 1 $, $ 0 < \alpha < 1 $ and $ w\in L^1_{loc}(\mathbb{R}^N) $, $ w\geq 0 $. Notice that in the limit case $ \alpha \to 1^- $, the above equation reduces to (1.1).

    (Q4) Same question as above for the problems

    $ u_{tt}-\Delta u +u_t = |u|^p+\frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}|\nabla u(s,x)|^{q}\,ds+w(x) $

    and

    $ u_{tt}-\Delta u +u_t = \frac{1}{\Gamma(1-\alpha)}\int_0^t (t-s)^{-\alpha}|u(s,x)|^{p}\,ds +\frac{1}{\Gamma(1-\beta)}\int_0^t(t-s)^{-\beta}|\nabla u(s,x)|^{q} +w(x), $

    where $ 0 < \alpha, \beta < 1 $.

    The author is supported by Researchers Supporting Project RSP-2020/4, King Saud University, Saudi Arabia, Riyadh.

    The author declares no conflict of interest.

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