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First and second critical exponents for an inhomogeneous damped wave equation with mixed nonlinearities

  • We investigate the Cauchy problem for the nonlinear damped wave equation uttΔu+ut=|u|p+|u|q+w(x), where N1, p,q>1, wL1loc(RN), w0 and w0. Namely, we first obtain the Fujita critical exponent for the considered problem. Next, we determine its second critical exponent in the sense of Lee and Ni. In particular, we show that the nonlinear gradient term |u|q induces a phenomenon of discontinuity of the Fujita critical exponent.

    Citation: Bessem Samet. First and second critical exponents for an inhomogeneous damped wave equation with mixed nonlinearities[J]. AIMS Mathematics, 2020, 5(6): 7055-7070. doi: 10.3934/math.2020452

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  • We investigate the Cauchy problem for the nonlinear damped wave equation uttΔu+ut=|u|p+|u|q+w(x), where N1, p,q>1, wL1loc(RN), w0 and w0. Namely, we first obtain the Fujita critical exponent for the considered problem. Next, we determine its second critical exponent in the sense of Lee and Ni. In particular, we show that the nonlinear gradient term |u|q induces a phenomenon of discontinuity of the Fujita critical exponent.


    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 N1, p,q>1, wL1loc(RN), w0 and w0. 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+2N and RNuj(x)dx>0, j=0,1, then the problem (1.2) and (1.3) admits no global solution;

    (ⅱ) if 1+2N<p<N for N3, and 1+2N<p< for N{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 (ttΔ+t)k. In [15], it was shown that the exponent 1+2N belongs to the blow-up case (ⅰ). Notice that 1+2N is also the Fujita critical exponent for the semilinear heat equation utΔu=|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, α0, and f(t,x)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 wL1loc(RN) and w0. It was shown that in the case N3, the critical exponent for the problem (1.5) jumps from 1+2N (the critical exponent for the problem (1.2)) to the bigger exponent 1+2N2. Notice that a similar phenomenon was observed for the inhomogeneous semilnear heat equation utΔu=|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 |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,)×RN. We denote by C2c(Q) the space of C2 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,u)Lploc(Q)×Lqloc(Q)

    and

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

    for all φC2c(Q).

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

    Theorem 1.1. Let N1, p,q>1, wL1loc(RN), w0 and w0.

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

    (ii) Let N3. If

    1<p<1+2N2or1<q<1+1N1,

    then problem (1.1) admits no global weak solution.

    (iii) Let N3. If

    p>1+2N2andq>1+1N1,

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

    We mention below some remarks related to Theorem 1.1. For N3, let

    pc(N,q)={1+2N2ifq>1+1N1,ifq<1+1N1.

    From Theorem 1.1, if 1<p<pc(N,q), then problem (1.1) admits no global weak solution, while if p>pc(N,q), global weak solutions exist for some w>0. This shows that pc(N,q) is the Fujita critical exponent for the problem (1.1). Notice that in the case N3 and q>1+1N1, pc(N,q) is also the critical exponent for uttΔu=|u|p+w(x) (see [5]). This shows that in the range q>1+1N1, the gradient term |u|q has no influence on the critical behavior of uttΔu=|u|p+w(x).

    It is interesting to note that the gradient term |u|q induces an interesting phenomenon of discontinuity of the Fujita critical exponent pc(N,q) jumping from 1+2N2 to as q reaches the value 1+1N1 from above.

    Next, for N3 and σ<N, we introduce the sets

    I+σ={wC(RN):w0,|x|σ=O(w(x)) as |x|}

    and

    Iσ={wC(RN):w>0,w(x)=O(|x|σ) as |x|}.

    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 N3, p>1+2N2 and q>1+1N1.

    (i) If

    σ<max{2pp1,qq1}

    and wI+σ, then problem (1.1) admits no global weak solution.

    (ii) If

    σmax{2pp1,qq1},

    then problem (1.1) admits global solutions for some wIσ.

    From Theorem 1.2, if N3, p>1+2N2 and q>1+1N1, then problem (1.1) admits a second critical exponent, namely

    σ=max{2pp1,qq1}.

    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,gC([0,)) satisfying

    f0,f0,supp(f)(0,1)

    and

    0g1,g(s)={1if0s1,0ifs2.

    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 2 and ρ>0 are constants to be chosen. It can be easily seen that

    ξT0andξTC2c(Q).

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

    Lemma 2.1. Let κ>1 and >2κκ1. 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 Tρ<|x|<2Tρ, an elementary calculation yields

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

    where

    θ(x)=Ng(r2T2ρ)g(r2T2ρ)+2(1)T2ρr2g(r2T2ρ)g(r2T2ρ)2+2T2ρr2g(r2T2ρ)g

    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

    \begin{eqnarray*} \int_{\mathbb{R}^N}|G|^{\frac{-1}{\kappa-1}}|\Delta G|^{\frac{\kappa}{\kappa-1}}\,dx &\leq & C T^{\frac{-2\rho \kappa}{\kappa-1}} \int_{T^\rho \lt |x| \lt \sqrt 2T^\rho} g\left(\frac{|x|^2}{T^{2\rho}}\right)^{\ell-\frac{2\kappa}{\kappa-1}}\,dx\\ & = & C T^{\frac{-2\rho \kappa}{\kappa-1}+N\rho} \int_{1 \lt |y| \lt \sqrt{2}}g(|y|^2)^{\ell-\frac{2\kappa}{\kappa-1}}\,dy, \end{eqnarray*}

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

    \begin{equation} \int_{\mathbb{R}^N}|G|^{\frac{-1}{\kappa-1}}|\Delta G|^{\frac{\kappa}{\kappa-1}}\,dx\leq CT^{\frac{-2\rho \kappa}{\kappa-1}+N\rho}. \end{equation} (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

    \begin{equation} \int_Q |\xi_T|^{\frac{-1}{\kappa-1}}|\nabla\xi_T|^{\frac{\kappa}{\kappa-1}}\,dx\,dt\leq C T^{\frac{-\rho \kappa}{\kappa-1}+N\rho+1}. \end{equation} (2.7)

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

    \begin{equation} \int_Q |\xi_T|^{\frac{-1}{\kappa-1}}|\nabla\xi_T|^{\frac{\kappa}{\kappa-1}}\,dx\,dt = \left(\int_0^\infty F(t)\,dt\right)\left(\int_{T^\rho \lt |x| \lt \sqrt 2 T^\rho}|G|^{\frac{-1}{\kappa-1}}|\nabla G|^{\frac{\kappa}{\kappa-1}}\,dx\right). \end{equation} (2.8)

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

    \begin{eqnarray*} |\nabla G(x)|& = &2\ell T^{-2\rho} |x| \left|g'\left(\frac{|x|^2}{T^{2\rho}}\right)\right|g\left(\frac{|x|^2}{T^{2\rho}}\right)^{\ell-1}\\ &\leq &2\sqrt{2}\ell T^{-\rho}\left|g'\left(\frac{|x|^2}{T^{2\rho}}\right)\right|g\left(\frac{|x|^2}{T^{2\rho}}\right)^{\ell-1}\\ &\leq & C T^{-\rho} g\left(\frac{|x|^2}{T^{2\rho}}\right)^{\ell-1}, \end{eqnarray*}

    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

    \begin{eqnarray*} \int_{T^\rho \lt |x| \lt \sqrt 2 T^\rho}|G|^{\frac{-1}{\kappa-1}}|\nabla G|^{\frac{\kappa}{\kappa-1}}\,dx&\leq & C T^{\frac{-\rho \kappa}{\kappa-1}} \int_{T^\rho \lt |x| \lt \sqrt 2 T^\rho}g\left(\frac{|x|^2}{T^{2\rho}}\right)^{\ell-\frac{\kappa}{\kappa-1}}\,dx\\ & = & C T^{\frac{-\rho \kappa}{\kappa-1}+N\rho}\int_{1 \lt |y| \lt \sqrt 2}g(|y|^2)^{\ell-\frac{\kappa}{\kappa-1}}\,dy, \end{eqnarray*}

    i.e.

    \begin{equation} \int_{T^\rho \lt |x| \lt \sqrt 2 T^\rho}|G|^{\frac{-1}{\kappa-1}}|\nabla G|^{\frac{\kappa}{\kappa-1}}\,dx \leq C T^{\frac{-\rho \kappa}{\kappa-1}+N\rho}. \end{equation} (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

    \begin{equation} \int_Q |\xi_T|^{\frac{-1}{\kappa-1}}|{\xi_T}_{tt}|^{\frac{\kappa}{\kappa-1}}\,dx\,dt \leq C T^{\frac{-2\kappa}{\kappa-1}+1+N\rho}. \end{equation} (2.10)

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

    \begin{equation} \int_Q |\xi_T|^{\frac{-1}{\kappa-1}}|{\xi_T}_{tt}|^{\frac{\kappa}{\kappa-1}}\,dx\,dt = \left(\int_0^T F(t)^{\frac{-1}{\kappa-1}}|F''(t)|^{\frac{\kappa}{\kappa-1}}\,dt\right)\left(\int_{0 \lt |x| \lt \sqrt 2 T^\rho} G(x)\,dx\right). \end{equation} (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} )

    \begin{equation} \int_0^T F(t)^{\frac{-1}{\kappa-1}}|F''(t)|^{\frac{\kappa}{\kappa-1}}\,dt\leq C T^{\frac{-2\kappa}{\kappa-1}+1}. \end{equation} (2.12)

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

    \begin{eqnarray*} \int_{0 \lt |x| \lt \sqrt 2 T^\rho} G(x)\,dx& = & \int_{0 \lt |x| \lt \sqrt 2 T^\rho} g\left(\frac{|x|^2}{T^{2\rho}}\right)^{\ell}\,dx\\ & = & T^{N\rho} \int_{0 \lt |y| \lt \sqrt 2} g(|y|^2)^\ell\,dy, \end{eqnarray*}

    which yields

    \begin{equation} \int_{0 \lt |x| \lt \sqrt 2 T^\rho} G(x)\,dx\leq C T^{N\rho}. \end{equation} (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

    \begin{equation} \int_Q |\xi_T|^{\frac{-1}{\kappa-1}}|{\xi_T}_{t}|^{\frac{\kappa}{\kappa-1}}\,dx\,dt \leq C T^{\frac{-\kappa}{\kappa-1}+1+N\rho}. \end{equation} (2.14)

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

    \begin{equation} \int_Q |\xi_T|^{\frac{-1}{\kappa-1}}|{\xi_T}_{t}|^{\frac{\kappa}{\kappa-1}}\,dx\,dt = \left(\int_0^T F(t)^{\frac{-1}{\kappa-1}}|F'(t)|^{\frac{\kappa}{\kappa-1}}\,dt\right)\left(\int_{0 \lt |x| \lt \sqrt 2 T^\rho} G(x)\,dx\right). \end{equation} (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} )

    \begin{equation} \int_0^T F(t)^{\frac{-1}{\kappa-1}}|F'(t)|^{\frac{\kappa}{\kappa-1}}\,dt\leq C T^{1-\frac{\kappa}{\kappa-1}}. \end{equation} (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

    \begin{equation} \int_Q w(x)\varphi\,dx\,dt \leq C \min\{A(\varphi),B(\varphi)\}, \end{equation} (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

    \begin{equation} \int_Q |u|^p\varphi\,dx\,dt+\int_Q w(x)\varphi\,dx\,dt \leq \int_Q |u||\varphi_{tt}|\,dx\,dt+\int_Q |u||\Delta \varphi|\,dx\,dt+\int_Q |u||\varphi_{t}|\,dx\,dt. \end{equation} (3.2)

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

    \begin{eqnarray} \int_Q |u||\varphi_{tt}|\,dx\,dt& = &\int_Q \left(|u|\varphi^{\frac{1}{p}}\right) \left(\varphi^{\frac{-1}{p}}|\varphi_{tt}|\right)\,dx\,dt\\ &\leq & \varepsilon \int_Q |u|^p\varphi\,dx\,dt+C \int_Q \varphi^{\frac{-1}{p-1}}|\varphi_{tt}|^{\frac{p}{p-1}}\,dx\,dt. \end{eqnarray} (3.3)

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

    \begin{equation} \int_Q |u||\Delta \varphi|\,dx\,dt\leq \varepsilon \int_Q |u|^p\varphi\,dx\,dt+C \int_Q \varphi^{\frac{-1}{p-1}}|\Delta \varphi|^{\frac{p}{p-1}}\,dx\,dt \end{equation} (3.4)

    and

    \begin{equation} \int_Q |u||\varphi_t|\,dx\,dt\leq \varepsilon \int_Q |u|^p\varphi\,dx\,dt+C \int_Q \varphi^{\frac{-1}{p-1}}|\varphi_t|^{\frac{p}{p-1}}\,dx\,dt \end{equation} (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

    \begin{equation} \int_Q w(x)\varphi\,dx\,dt\leq C A(\varphi). \end{equation} (3.6)

    Again, by (1.6), one has

    \begin{aligned} &\int_Q \left(|u|^p+|\nabla u|^q\right)\varphi\,dx\,dt +\int_{Q} w(x)\varphi \,\,dx\,dt\\ & = \int_Q u\varphi_{tt}\,dx\,dt -\int_Q u \Delta\varphi\,dx\,dt -\int_Q u \varphi_t\,dx\,dt\\ & = \int_Q u\varphi_{tt}\,dx\,dt +\int_Q \nabla u \cdot \nabla \varphi\,dx\,dt -\int_Q u \varphi_t\,dx\,dt, \end{aligned}

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

    \begin{equation} \begin{aligned} &\int_Q \left(|u|^p+|\nabla u|^q\right)\varphi\,dx\,dt +\int_{Q} w(x)\varphi \,\,dx\,dt\\ &\leq \int_Q |u||\varphi_{tt}|\,dx\,dt+\int_Q |\nabla u| |\nabla \varphi|\,dx\,dt +\int_Q |u| |\varphi_t|\,dx\,dt. \end{aligned} \end{equation} (3.7)

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

    \begin{eqnarray} \int_Q |u||\varphi_{tt}|\,dx\,dt&\leq & \frac{1}{2} \int_Q |u|^p\varphi\,dx\,dt+C \int_Q \varphi^{\frac{-1}{p-1}}|\varphi_{tt}|^{\frac{p}{p-1}}\,dx\,dt, \end{eqnarray} (3.8)
    \begin{eqnarray} \int_Q |u||\varphi_{t}|\,dx\,dt&\leq & \frac{1}{2} \int_Q |u|^p\varphi\,dx\,dt+C \int_Q \varphi^{\frac{-1}{p-1}}|\varphi_{t}|^{\frac{p}{p-1}}\,dx\,dt, \end{eqnarray} (3.9)
    \begin{eqnarray} \int_Q |\nabla u| |\nabla \varphi|\,dx\,dt &\leq & \frac{1}{2} \int_Q |\nabla u|^q\varphi\,dx\,dt+C\int_Q \varphi^{\frac{-1}{q-1}}|\nabla \varphi|^{\frac{q}{q-1}}\,dx\,dt. \end{eqnarray} (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

    \begin{equation} \int_{Q} w(x)\varphi \,\,dx\,dt\leq CB(\varphi). \end{equation} (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

    \begin{equation} \int_Q w(x)\xi_T\,dx\,dt \leq C \min\{A(\xi_T),B(\xi_T)\}, \end{equation} (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 ,

    \begin{equation} \int_Q w(x)\xi_T\,dx\,dt\geq CT, \end{equation} (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

    \begin{equation} \int_Q w(x)\xi_T\,dx\,dt = \left(\int_0^T f\left(\frac{t}{T}\right)^\ell \,dt\right) \left(\int_{0 \lt |x| \lt \sqrt{2}T^\rho} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx\right). \end{equation} (3.14)

    On the other hand,

    \begin{equation} \int_0^T f\left(\frac{t}{T}\right)^\ell \,dt = T \int_0^1 f(s)^\ell\,ds. \end{equation} (3.15)

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

    \begin{equation} \int_{0 \lt |x| \lt \sqrt{2}T^\rho} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx\geq \int_{0 \lt |x| \lt 1} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx. \end{equation} (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 ,

    \begin{equation} \int_{0 \lt |x| \lt 1} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx\geq C. \end{equation} (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

    \begin{equation} CT\leq \min\{A(\xi_T),B(\xi_T)\}, \end{equation} (3.18)

    where \xi_T is defined by (2.1),

    \begin{equation} A(\xi_T) = \int_Q \xi_T^{\frac{-1}{p-1}}|{\xi_T}_{tt}|^{\frac{p}{p-1}}\,dx\,dt+\int_Q \xi_T^{\frac{-1}{p-1}}|{\xi_T}_{t}|^{\frac{p}{p-1}}\,dx\,dt +\int_Q \xi_T^{\frac{-1}{p-1}}|\Delta \xi_T|^{\frac{p}{p-1}}\,dx\,dt \end{equation} (3.19)

    and

    \begin{equation} B(\xi_T) = \int_Q \varphi^{\frac{-1}{p-1}}|{\xi_T}_{tt}|^{\frac{p}{p-1}}\,dx\,dt+\int_Q \xi_T^{\frac{-1}{p-1}}|{\xi_T}_{t}|^{\frac{p}{p-1}}\,dx\,dt +\int_Q \xi_T^{\frac{-1}{q-1}}|\nabla \xi_T|^{\frac{q}{q-1}}\,dx\,dt. \end{equation} (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

    \begin{eqnarray} \int_Q \xi_T^{\frac{-1}{p-1}}|{\xi_T}_{tt}|^{\frac{p}{p-1}}\,dx\,dt &\leq & C T^{\frac{-2p}{p-1}+1+N\rho}, \end{eqnarray} (3.21)
    \begin{eqnarray} \int_Q \xi_T^{\frac{-1}{p-1}}|{\xi_T}_{t}|^{\frac{p}{p-1}}\,dx\,dt &\leq & C T^{\frac{-p}{p-1}+1+N\rho}, \end{eqnarray} (3.22)
    \begin{eqnarray} \int_Q \xi_T^{\frac{-1}{p-1}}|\Delta \xi_T|^{\frac{p}{p-1}}\,dx\,dt &\leq & C T^{\frac{-2\rho p}{p-1}+N\rho+1}, \end{eqnarray} (3.23)
    \begin{eqnarray} \int_Q \xi_T^{\frac{-1}{q-1}}|\nabla \xi_T|^{\frac{q}{q-1}}\,dx\,dt &\leq & C T^{\frac{-\rho q}{q-1}+N\rho+1}. \end{eqnarray} (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

    \begin{equation} \frac{-2p}{p-1}+1+N\rho \lt \frac{-p}{p-1}+1+N\rho. \end{equation} (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

    \begin{equation} A(\xi_T)\leq C T^{\frac{-p}{p-1}+1+\frac{N}{2}} \end{equation} (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

    \begin{equation} B(\xi_T)\leq C T^{\frac{-p}{p-1}+1+\frac{Np(q-1)}{q(p-1)}}. \end{equation} (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

    \begin{equation} 0 \lt C\leq T^{\frac{-p}{p-1}+\frac{N}{2}}: = T^{\alpha(N)} \end{equation} (3.28)

    and

    \begin{equation} 0 \lt C\leq T^{\frac{-p}{p-1}+\frac{Np(q-1)}{q(p-1)}}: = T^{\beta(N)}. \end{equation} (3.29)

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

    \alpha(N) = \left\{\begin{array}{lll} \frac{-p-1}{2(p-1)} &\mbox{if}& N = 1,\\ \frac{-1}{p-1} &\mbox{if}& N = 2, \end{array} \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

    \begin{equation} p \gt 1+\frac{2}{N-2}\quad\mbox{and}\quad q \gt 1+\frac{1}{N-1}. \end{equation} (3.30)

    Let

    \begin{equation} u(x) = \varepsilon (1+r^2)^{-\delta},\quad r = |x|,\quad x\in \mathbb{R}^N, \end{equation} (3.31)

    where

    \begin{equation} \max\left\{\frac{1}{p-1},\frac{2-q}{2(q-1)}\right\} \lt \delta \lt \frac{N-2}{2} \end{equation} (3.32)

    and

    \begin{equation} 0 \lt \varepsilon \lt \min\left\{1,\left(\frac{2\delta(N-2\delta-2)}{1+2^q\delta^q}\right)^{\frac{1}{\min\{p,q\}-1}}\right\}. \end{equation} (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

    \begin{eqnarray} w(x)& = &2\delta \varepsilon \left[(1+r^2)^{-\delta-1}-2r^2(\delta+1)(1+r^2)^{-\delta-2}+(N-1)(1+r^2)^{-\delta-1}\right]\\ && -\varepsilon^p(1+r^2)^{-\delta p}-2^q\delta^q\varepsilon^qr^q(1+r^2)^{-(\delta+1)q}. \end{eqnarray} (3.34)

    Hence, one obtains

    \begin{eqnarray*} w(x) &\geq & 2\delta \varepsilon \left[(1+r^2)^{-\delta-1}-2(\delta+1)(1+r^2)^{-\delta-1}+(N-1)(1+r^2)^{-\delta-1}\right]\\ &&-\varepsilon^p(1+r^2)^{-\delta p}-2^q\delta^q\varepsilon^q(1+r^2)^{-(\delta+1)q+\frac{q}{2}}\\ & = & 2\delta \varepsilon (N-2\delta-2)(1+r^2)^{-\delta-1}-\varepsilon^p(1+r^2)^{-\delta p}-2^q\varepsilon^q\delta^q(1+r^2)^{-\left(\delta+\frac{1}{2}\right)q}. \end{eqnarray*}

    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 ,

    \begin{equation} \int_Q w(x)\xi_T\,dx\,dt \geq C T^{\rho(N-\sigma)+1}. \end{equation} (4.1)

    where \xi_T is defined by (2.1).

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

    \begin{equation} \int_Q w(x)\xi_T\,dx\,dt = CT\int_{0 \lt |x| \lt \sqrt{2}T^\rho} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx. \end{equation} (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 )

    \begin{eqnarray*} \int_{0 \lt |x| \lt \sqrt{2}T^\rho} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx &\geq & \int_{0 \lt |x| \lt T^\rho} w(x) g\left(\frac{|x|^2}{T^{2\rho}}\right)^\ell\,dx\\ & = & \int_{0 \lt |x| \lt T^\rho} w(x)\,dx\\ &\geq & \int_{\frac{T^\rho}{2} \lt |x| \lt T^\rho} w(x)\,dx\\ &\geq & C \int_{\frac{T^\rho}{2} \lt |x| \lt T^\rho} |x|^{-\sigma}\,dx \\ & = & C T^{\rho(N-\sigma)}. \end{eqnarray*}

    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

    \begin{equation} CT^{\rho(N-\sigma)+1}\leq \min\{A(\xi_T),B(\xi_T)\}, \end{equation} (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

    \begin{equation} 0 \lt C\leq T^{\frac{-p}{p-1}+\frac{\sigma}{2}}. \end{equation} (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

    \begin{equation} 0 \lt C\leq T^{\frac{p}{p-1}\left(\frac{\sigma(q-1)}{q}-1\right)}. \end{equation} (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

    \begin{equation} \sigma \geq \max\left\{\frac{2p}{p-1},\frac{q}{q-1}\right\}. \end{equation} (4.6)

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

    \begin{equation} \frac{\sigma-2}{2} \lt \delta \lt \frac{N-2}{2} \end{equation} (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\{\begin{array}{lll} 1+\frac{2}{N-2} &\mbox{if}& q \gt 1+\frac{1}{N-1},\\ \infty &\mbox{if}& q \lt 1+\frac{1}{N-1}. \end{array} \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

    \begin{eqnarray*} \left\{\begin{array}{lllll} u_{tt}-\Delta u +u_t & = & |v|^{p_1}+|\nabla v|^{q_1} +w_1(x) &\mbox{in}& (0,\infty)\times \mathbb{R}^N,\\ v_{tt}-\Delta v +v_t& = & |u|^{p_2}+|\nabla u|^{q_2} +w_2(x) &\mbox{in}& (0,\infty)\times \mathbb{R}^N, \end{array} \right. \end{eqnarray*}

    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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