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Editorial Special Issues

Autism and neuro-immune-gut link

  • Received: 28 February 2018 Accepted: 20 June 2018 Published: 27 June 2018
  • Recent evidences sustain the hypothesis that host-bacteria interactions play a critical role in regulating tissue and body homeostasis. Gut microbiota and the brain are strongly interconnected and share communication pathways. Modifications in gut bacteria compositions are correlated to changes in behaviors. Indeed, autism spectrum disorders (ASD) are linked to dysfunctions of the gut bacteria-brain axis. Possible therapeutic strategies in ASD management will aim to restore dysbiosis and gut bacteria imbalance.

    Citation: Dario Siniscalco, Anna Lisa Brigida, Nicola Antonucci. Autism and neuro-immune-gut link[J]. AIMS Molecular Science, 2018, 5(2): 166-172. doi: 10.3934/molsci.2018.2.166

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    In this paper, we study the initial boundary value problem of the nonlinear viscoelastic hyperbolic problem with variable exponents:

    $ {utt+2u+2uttt0g(tτ)2u(τ)dτ+|ut|m(x)2ut=|u|p(x)2u,(x,t)Ω×(0,T),u(x,t)=uν(x,t)=0,(x,t)Ω×(0,T),u(x,0)=u0(x), ut(x,0)=u1(x),xΩ,
    $
    (1.1)

    where $ \Omega\subset R^{n}(n\geq1) $ is a bounded domain in $ R^n $ with a smooth boundary $ \partial \Omega $, $ \nu $ is the unit outer normal to $ \partial\Omega $, the exponents $ m(x) $ and $ p(x) $ are continuous functions on $ \overline{\Omega} $ with the logarithmic module of continuity:

    $ x,yΩ,|xy|<1,|m(x)m(y)|+|p(x)p(y)|ω(|xy|),
    $
    (1.2)

    where

    $ limτ0+supω(τ)ln1τ=C<.
    $
    (1.3)

    In addition to this condition, the exponents satisfy the following:

    $ 2m:=essinfxΩm(x)m(x)m+:=esssupxΩm(x)<2(n2)n4,
    $
    (1.4)
    $ 2p:=essinfxΩp(x)p(x)p+:=esssupxΩp(x)<2(n2)n4,
    $
    (1.5)

    $ g: R^{+} \, \rightarrow R^{+} $ is a $ C^{1} $ function satisfying

    $ g(0)>0, g(τ)0,10g(τ)dτ=l>0.
    $
    (1.6)

    The equation of Problem $ (1.1) $ arises from the modeling of various physical phenomena such as the viscoelasticity and the system governing the longitudinal motion of a viscoelastic configuration obeying a nonlinear Boltzmann's model, or electro-rheological fluids, viscoelastic fluids, processes of filtration through a porous medium, and fluids with temperature-dependent viscosity and image processing which give rise to equations with nonstandard growth conditions, that is, equations with variable exponents of nonlinearities. More details on these problems can be found in previous studies [1,2,3,4,5,6].

    When $ m(x) $ and $ p(x) $ are constants, Messaoudi [7] discussed the nonlinear viscoelastic wave equation

    $ uttu+t0g(tτ)u(τ)dτ+|ut|m2ut=|u|p2u,
    $

    he proved that any weak solution with negative initial energy blows up in finite time if $ p > m $, and a global existence result for $ p\leq m $. The results were improved later by Messaoudi [8], where the blow-up result in finite time with positive initial energy was obtained. Moreover, Song [9] showed the finite-time blow-up of some solutions whose initial data had arbitrarily high initial energy. In the same year, Song [10] studied the initial-boundary value problem

    $ |ut|ρuttu+t0g(tτ)u(τ)dτ+|ut|m2ut=|u|p2u,
    $

    and proved the nonexistence of global solutions with positive initial energy. Cavalcanti, Domingos, and Ferreira [11] were concerned with the non-linear viscoelastic equation

    $ |ut|ρuttuutt+t0g(tτ)u(τ)dτγut=0,
    $

    and proved the global existence of weak solutions. Moreover, they obtained the uniform decay rates of the energy by assuming a strong damping $ \triangle u_{t} $ acting in the domain and providing the relaxation function which decays exponentially.

    In 2017, Messaoudi [12] considered the following nonlinear wave equation with variable exponents:

    $ uttu+a|ut|m(x)2ut=b|u|p(x)2u,
    $

    where $ a, b $ are positive constants. By using the Faedo$ - $Galerkin method, the existence of a unique weak solution is established under suitable assumptions on the variable exponents $ m(x) $ and $ p(x) $. Then this paper also proved the finite-time blow-up of solutions and gave a two-dimensional numerical example to illustrate the blow up result. Park [13] showed the blow up of solutions for a viscoelastic wave equation with variable exponents

    $ uttu+t0g(ts)u(s)ds+a|ut|m(x)2ut=b|u|p(x)2u,
    $

    where the exponents of nonlinearity $ p(x) $ and $ m(x) $ are given functions and $ a, b > 0 $ are constants. For nonincreasing positive function $ g $, they prove the blow-up result for the solutions with positive initial energy as well as nonpositive initial energy. Alahyane [14] discussed the nonlinear viscoelastic wave equation with variable exponents

    $ uttu+t0g(tτ)u(τ)dτ+μut=|u|p(x)2u,
    $

    where $ \mu $ is a nonnegative constant and the exponent of nonlinearity $ p(x) $ and $ g $ are given functions. Under arbitrary positive initial energy and specific conditions on the relaxation function $ g $, they prove a finite-time blow-up result and give some numerical applications to illustrate their theoretical results. Ouaoua and Boughamsa [15] considered the following boundary value problem:

    $ utt+2uu+|ut|m(x)2ut=|u|p(x)2u,
    $

    the authors established the local existence by using the Faedo$ - $Galerkin method with positive initial energy and suitable conditions on the variable exponents $ m(x) $ and $ r(x) $. In addition, they also proved that the local solution is global and obtained the stability estimate of the solution. Ding and Zhou [16] considered a Timoshenko-type equation

    $ utt+2uM(||u||22)u+|ut|p(x)2ut=|u|q(x)2u,
    $

    they prove that the solutions blow up in finite time with positive initial energy. Therefore, the existence of finite-time blow-up solutions with arbitrarily high initial energy is established, and the upper and lower bounds of the blow-up time are derived. More related references can be found in [17,18,19,20,21,22].

    Motivated by [7,13,14], we considered the existence of the solutions and their blow-up for the nonlinear damping and viscoelastic hyperbolic problem with variable exponents. Our aim in this work is to prove the existence of the weak solutions and to find sufficient conditions on $ m(x) $ and $ p(x) $ for which the blow-up takes place.

    This article consists of three sections in addition to the introduction. In Section 2, we recall the definitions and properties of $ L^{p(x)}(\Omega) $ and the Sobolev spaces $ W^{1, p(x)}(\Omega) $. In Section 3, we prove the existence of weak solutions for Problem (1.1). In Section 4, we state and prove the blow-up result for solutions with positive initial energy as well as nonpositive initial energy.

    In this section, we review some results regarding Lebesgue and Sobolev spaces with variable exponents first. All of these results and a comprehensive study of these spaces can be found in [23]. Here $ (\cdot, \cdot) $ and $ \langle \cdot, \cdot \rangle $ denote the inner product in space $ L^{2}(\Omega) $ and the duality pairing between $ H^{-2}(\Omega) $ and $ H_{0}^{2}(\Omega) $.

    The variable exponent Lebesgue space $ L^{p(x)}(\Omega) $ is defined by

    $ L^{p(x)}(\Omega) = \left\{u(x): u\, \, {\rm is\, \, \rm measurable\, \, \rm in}\, \, \Omega, \ \rho_{p(x)}(u) = \int_{\Omega}|u|^{p(x)}dx < \infty\right\}, $

    this space is endowed with the norm

    $ \|u\|_{p(x)} = \mbox{inf}\ \left\{\lambda > 0:\int_{\Omega}\Big|\frac{u(x)}{\lambda}\Big|^{p(x)}\mathrm{d}x\leq1\right\}. $

    The variable exponent Sobolev space $ W^{1, p(x)}(\Omega) $ is defined by

    $ W^{1, p(x)}(\Omega) = \left\{u\in L^{p(x)}(\Omega)\ \, {\rm such \, \, \rm that}\, \, \nabla u \, \, {\rm exists\, \, \rm and}\, \, |\nabla u|\in L^{p(x)}(\Omega)\right\}, $

    the corresponding norm for this space is

    $ \|u\|_{1, p(x)} = \|u\|_{{p(x)}}+\|\nabla u\|_{{p(x)}}, $

    define$ \ W_0^{1, p(x)}(\Omega) $ as the closure of $ \ C_0^\infty(\Omega) $ with respect to the$ \ W^{1, p(x)}(\Omega) $ norm. The spaces $ \ L^{p(x)}(\Omega), W^{1, p(x)}(\Omega) $ and $ W_0^{1, p(x)}(\Omega) $ are separable and reflexive Banach spaces when $ 1 < p^-\leq p^+ < \infty $, where $ p^-: = ess\inf\limits_{\Omega} p(x) $ and $ p^+: = ess\sup\limits_{\Omega} p(x). $ As usual, we denote the conjugate exponent of $ p(x) $ by $ p'(x) = p(x)/(p(x)-1) $ and the Sobolev exponent by

    $ p^*(x) = \left\{np(x)nkp(x),if p(x)<n,,if p(x)n.
    \right. $

    Lemma 2.1. If $ p_{1}(x), \ p_{2}(x)\in C_{+}(\overline{\Omega}) = \{h\in C(\overline{\Omega}):\min\limits_{x\in \overline{\Omega}} h(x) > 1\} $, $ p_{1}(x)\leq p_{2}(x) $ for any $ x\in\Omega $, then there exists the continuous embedding $ L^{p_{2}(x)}(\Omega)\hookrightarrow L^{p_{1}(x)}(\Omega) $, whose norm does not exceed $ |\Omega|+1 $.

    Lemma 2.2. Let $ p(x), \ q(x)\in C_{+}(\overline{\Omega}). $ Assuming that $ q(x) < p^{\ast}(x) $, there is a compact and continuous embedding $ W^{k, p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega). $

    Lemma 2.3. (Hölder's inequality) [24] For any $ u\in L^{p(x)}(\Omega) $ and $ v\in L^{q(x)}(\Omega) $, then the following inequality holds:

    $ \left|\int_{\Omega}uvdx\right|\leq(\frac{1}{p^{-}}+\frac{1}{q^{-}})||u||_{p(x)}||v||_{q(x)}\leq2||u||_{p(x)}||v||_{q(x)}. $

    Lemma 2.4. For $ u\in L^{p(x)}(\Omega) $, the following relations hold:

    $ u\neq0 \Rightarrow \Big(\|u\|_{p(x)} = \lambda\Leftrightarrow\rho_{p(x)}(\frac{u}{\lambda}) = 1\Big ), $
    $ \|u\|_{p(x)} < 1( = 1; > 1)\Leftrightarrow\rho_{p(x)}(u) < 1( = 1; > 1), $
    $ \|u\|_{p(x)} > 1\Rightarrow \|u\|_{p(x)}^{p^{-}}\leq\rho_{p(x)}(u)\leq\|u\|_{p(x)}^{p^+}, $
    $ \|u\|_{p(x)} < 1\Rightarrow \|u\|_{p(x)}^{p^+}\leq\rho_{p(x)}(u)\leq\|u\|_{p(x)}^{p^-}. $

    Next, we give the definition of the weak solution to Problem $ (1.1) $.

    Definition 2.1. A function u(x, t) is called a weak solution for Problem $ (1.1) $, if $ u\in C(0, T;H_{0}^{2}(\Omega)) $ $ \cap C^{1}(0, T;H_{0}^{2}(\Omega))\cap C^{2}(0, T;H^{-2}(\Omega)) $ with $ u_{tt}\in L^{2}(0, T;H_{0}^{2}(\Omega)) $ and u satisfies the following conditions:

    (1) For every $ \omega \in H_{0}^{2}(\Omega) $ and for $ a.e. \, t\in (0, T) $

    $ \langle u_{tt}, \omega\rangle+(\triangle u, \triangle \omega)+(\triangle u_{tt}, \triangle \omega)-\int_{0}^{t}g(t-\tau)(\triangle u(\tau), \triangle \omega)d\tau\\+(|u_{t}|^{m(x)-2}u_{t}, \omega) = (|u|^{p(x)-2}u, \omega), $

    (2) $ u(x, 0) = u_{0}(x)\in H_{0}^{2}(\Omega), \, u_{t}(x, 0) = u_{1}(x)\in H_{0}^{2}(\Omega). $

    In this section, we prove the existence of a weak solution for Problem $ (1.1) $ by making use of the Faedo–Galerkin method and the contraction mapping principle. For a fixed $ T > 0 $, we consider the space $ \mathscr H = C(0, T;H_{0}^{2}(\Omega))\cap C^{1}(0, T;H_{0}^{2}(\Omega)) $ with the norm $ ||v||_{\mathscr H}^{2} = \max\limits_{0\leq t\leq T} (||\triangle v_{t}||_{2}^{2}+l||\triangle v||_{2}^{2}) $.

    Lemma 3.1. Assume that $ (1.4) $, $ (1.5) $, and $ (1.6) $ hold, let $ (u_{0}, u_{1})\in H_{0}^{2}(\Omega)\times H_{0}^{2}(\Omega) $, for any $ T > 0 $, $ v\in \mathscr H $, then there exists $ u\in C(0, T;H_{0}^{2}(\Omega))\cap C^{1}(0, T;H_{0}^{2}(\Omega))\cap C^{2}(0, T;H^{-2}(\Omega)) $ with $ \ u_{tt}\in L^{2}(0, T;H_{0}^{2}(\Omega)) $ satisfying

    $ {utt+2u+2uttt0g(tτ)2u(τ)dτ+|ut|m(x)2ut=|v|p(x)2v,(x,t)Ω×(0,T),u(x,t)=uν(x,t)=0,(x,t)Ω×(0,T),u(x,0)=u0(x), ut(x,0)=u1(x),xΩ.
    $
    (3.1)

    Proof. Let $ \{\omega_{j}\}_{j = 1}^{\infty} $ be the orthogonal basis of $ H\mathcal{}_{0}^{2}(\Omega) $, which is the standard orthogonal basis in $ L^{2}(\Omega) $ such that

    $ -\triangle\omega_{j} = \lambda_{j}\omega_{j} \ \ \rm in\ \ \Omega, \, \, \omega_{j} = 0 \ \ \rm on\ \ \partial \Omega, $

    we denote by $ V_{k} = {\rm span}\{\omega_{1}, \omega_{2}, \cdot\cdot\cdot, \omega_{k}\} $ the subspace generated by the first $ k $ vectors of the basis $ \{\omega_{j}\}_{j = 1}^{\infty} $. By normalization, we have $ ||\omega_{j}||_{2} = 1 $. For all $ k\geq1 $, we seek $ k $ functions $ c_{1}^{k}(t), c_{2}^{k}(t), \ldots, c_{k}^{k}(t)\in C^{2}[0, T] $ such that

    $ u^{k}(x, t) = \sum\limits_{j = 1}^{k}c_{j}^{k}(t)\omega_{j}(x), $

    satisfying the following approximate problem

    $ {(uktt,ωi)+(uk,ωi)+(uktt,ωi)t0g(tτ)(uk,ωi)dτ+(|ukt|m(x)2ukt,ωi)=Ω|v|p(x)2vωidx,uk(0)=uk0,   ukt(0)=uk1,     i=1,2,k,
    $
    (3.2)

    where

    $ u_{0}^{k} = \sum\limits_{i = 1}^{k}(u_{0}, \omega_{i})\omega_{i}\rightarrow u_{0}\ \ \ \rm in\ \ H_{0}^{2}(\Omega), $
    $ u_{1}^{k} = \sum\limits_{i = 1}^{k}(u_{1}, \omega_{i})\omega_{i}\rightarrow u_{1}\ \ \ \rm in\ \ H_{0}^{2}(\Omega), $

    thus, $ (3.2) $ generates the initial value problem for the system of second-order differential equations with respect to $ c_{i}^{k}(t) $:

    $ {(1+λ2i)ckitt(t)+λ2icki(t)=Gi(ck1t(t),,ckkt(t))+gi(cki(t)),   i=1,2,,k,cki(0)=Ωu0ωidx,      ckit(0)=Ωu1ωidx,              i=1,2,,k.
    $
    (3.3)

    where

    $ Gi(ck1t(t),,ckkt(t))=Ω|kj=1ckjt(t)ωj(x)|m(x)2kj=1ckjt(t)ωj(x)ωi(x)dx,
    $

    and

    $ gi(cki(t))=λ2it0g(tτ)cki(τ)dτ+Ω|v|p(x)2vωidx,
    $

    by Peano's Theorem, we infer that the Problem $ (3.3) $ admits a local solution $ c_{i}^{k}(t)\in C^{2}[0, T] $.

    $ \textbf{The} $ $ \textbf{first} $ $ \textbf{estimate}. $ Multiplying $ (3.2) $ by $ c_{it}^{k}(t) $ and summing with respect to $ i $, we arrive at the relation

    $ ddt(12||ukt||22+12||uk||22+12||ukt||22)+Ω|ukt|m(x)dxt0g(tτ)Ωuk(τ)uktdxdτ=Ω|v|p(x)2vuktdx.
    $
    (3.4)

    By simple calculation, we have

    $ t0g(tτ)ΩΔuk(τ)Δuktdxdτ=12ddt(guk)12(guk)12ddtt0g(τ)dτ||Δuk||22+12g(t)||uk||22,
    $
    (3.5)

    where

    $ (\varphi\diamond\bigtriangleup\psi) = \int_{0}^{t}\varphi(t-\tau)||\triangle\psi(t)-\triangle\psi(\tau)||_{2}^{2}d\tau, $

    inserting $ (3.5) $ into $ (3.4) $, using Hölder's inequality and Young's inequality, we obtain

    $ ddt[12||ukt||22+12||ukt||22+12(guk)+12(1t0g(τ)dτ)||Δuk||22]=12(guk)12g(t)||uk||22+Ω|v|p(x)2vuktdxΩ|ukt|m(x)dxΩ|v|p(x)2vuktdx|v|p(x)2v2||ukt||2η2Ω|v|2(p(x)1)dx+12η||ukt||22,
    $
    (3.6)

    using the embedding $ H_{0}^{2}(\Omega)\hookrightarrow L^{2(p(x)-1)}(\Omega) $ and Lemma $ 2.4 $, we easily obtain

    $ Ω|v|2(p(x)1)dxmax{||v||2(p1)2(p(x)1),||v||2(p+1)2(p(x)1)}Cmax{||v||2(p1)2,||v||2(p+1)2}C,
    $
    (3.7)

    where $ C $ is a positive constant. We denote by $ C $ various positive constants that may be different at different occurrences.

    Combining $ (3.6) $ and $ (3.7) $, we obtain

    $ ddt[12||ukt||22+12||ukt||22+12(guk)+12(1t0g(τ)dτ)||Δuk||22]η2C+12η||ukt||22,
    $

    by Gronwall's inequality, there exists a positive constant $ C_{T} $ such that

    $ ||ukt||22+||ukt||22+(guk)+l||Δuk||22CT,
    $
    (3.8)

    therefore, there exists a subsequence of $ \{u^{k}\}_{k = 1}^{\infty} $, which we still denote by $ \{u^{k}\}_{k = 1}^{\infty} $, such that

    $ uku weakly star in L(0,T;H20(Ω)),uktut weakly star in L(0,T;H20(Ω)),uku weakly in L2(0,T;H20(Ω)),uktut weakly in L2(0,T;H20(Ω)).
    $
    (3.9)

    $ \textbf{The} $ $ \textbf{second} $ $ \textbf{estimate}. $ Multiplying $ (3.2) $ by $ c_{itt}^{k}(t) $ and summing with respect to $ i $, we obtain

    $ ||uktt||22+||Δuktt||22+ddt(Ω1m(x)|ukt|m(x)dx)=Ωukukttdx+t0g(tτ)ΩΔuk(τ)Δukttdxdτ+Ω|v|p(x)2vukttdx.
    $
    (3.10)

    Note that we have the estimates for $ \varepsilon > 0 $

    $ |Ωukukttdx|ε||uktt||22+14ε||uk||22,
    $
    (3.11)
    $ Ω|v|p(x)2vukttdx|v|p(x)2v2uktt2ε||uktt||22+14εΩ|v|2(p(x)1)dx,
    $
    (3.12)

    and

    $ |t0g(tτ)Ωuk(τ)ukttdxdτ|14εΩ(t0g(tτ)uk(τ)dτ)2dx+ε||uktt||22ε||uktt||22+14εt0g(s)dst0g(tτ)Ω|uk(τ)|2dxdτε||uktt||22+(1l)g(0)4εt0||uk(τ)||22dτ,
    $
    (3.13)

    similar to $ (3.6) $ and $ (3.7) $, from $ H_{0}^{2}(\Omega)\hookrightarrow L^{2}(\Omega) $, we have

    $ Ω|v|p(x)2vukttdxεC||uktt||22+C4ε.
    $
    (3.14)

    Taking into account $ (3.10)-(3.14) $, we obtain

    $ ||uktt||22+(12εCε)||uktt||22+ddt(Ω1m(x)|ukt|m(x)dx)14ε||uk||22+(1l)g(0)4εt0||uk(τ)||22dτ+C4ε,
    $
    (3.15)

    integrating $ (3.15) $ over $ (0, t) $, we obtain

    $ t0||uktt||22dτ+(12εCε)t0||uktt||22dτ+Ω1m(x)|ukt|m(x)dxC4εt0(||uk||22+τ0||uk(s)||22ds)dτ+CT,
    $
    (3.16)

    taking $ \varepsilon $ small enough in (3.16), for some positive constant $ C_{T} $, we obtain

    $ t0||uktt||22dτ+t0||uktt||22dτCT,
    $
    (3.17)

    we observe that estimate $ (3.17) $ implies that there exists a subsequence of $ \{u^{k}\}_{k = 1}^{\infty} $, which we still denote by $ \{u^{k}\}_{k = 1}^{\infty} $, such that

    $ ukttutt weakly in L2(0,T;H20(Ω)).
    $
    (3.18)

    In addition, from $ (3.9) $, we have

    $ (uktt,ωi)=ddt(ukt,ωi)ddt(ut,ωi)=(utt,ωi)  weakly  star  in  L(0,T;H2(Ω)).
    $
    (3.19)

    Next, we will deal with the nonlinear term. Combining $ (3.9) $, $ (3.18) $, and Aubin–Lions theorem [25], we deduce that there exists a subsequence of $ \{u^{k}\}_{k = 1}^{\infty} $ such that

    $ uktut strongly in C(0,T;L2(Ω)),
    $
    (3.20)

    then

    $ |ukt|m(x)2ukt|ut|m(x)2ut  a.e. (x,t)Ω×(0,T),
    $
    (3.21)

    using the embedding $ H_{0}^{2}(\Omega)\hookrightarrow L^{2(m(x)-1)}(\Omega) $ and Lemma $ 2.4 $, we have

    $ |ukt|m(x)2ukt22=Ω|ukt|2(m(x)1)dxmax{||ukt||2(m1)2,||ukt||2(m+1)2}C,
    $
    (3.22)

    hence, using $ (3.21) $ and $ (3.22) $, we obtain

    $ |ukt|m(x)2ukt|ut|m(x)2ut  weakly  star in L(0,T;L2(Ω)).
    $
    (3.23)

    Setting up $ k\rightarrow \infty $ in $ (3.2) $, combining with $ (3.9) $, $ (3.18) $, $ (3.19) $, and $ (3.23) $, we obtain

    $ \langle u_{tt}, \omega \rangle+(\triangle u, \triangle \omega)+(\triangle u_{tt}, \triangle \omega)-\int_{0}^{t}g(t-\tau)(\triangle u(\tau), \triangle \omega)d\tau+(|u_{t}|^{m(x)-2}u_{t}, \omega) = (|v|^{p(x)-2}v, \omega). $

    To handle the initial conditions. From $ (3.9) $ and Aubin–Lions theorem, we can easily get $ u^{k}\rightarrow u $ in $ C(0, T;L^{2}(\Omega)) $, thus $ u^{k}(0)\rightarrow u(0) $ in $ L^{2}(\Omega) $, and we also have that $ u^{k}(0) = u_{0}^{k}\rightarrow u_{0} $ in $ H_{0}^{2}(\Omega) $, hence $ u(0) = u_{0} $ in $ H_{0}^{2}(\Omega) $. Similarly, we get that $ u_{t}(0) = u_{1} $.

    Uniqueness. Suppose that $ (3.1) $ has solutions $ u $ and $ z $, then $ \omega = u-z $ satisfies

    $ {ωtt+2ω+2ωttt0g(tτ)2ω(τ)dτ+|ut|m(x)2ut|zt|m(x)2zt=0,(x,t)Ω×(0,T),ω(x,t)=ων(x,t)=0,(x,t)Ω×(0,T),ω(x,0)=0, ωt(x,0)=0,xΩ.
    $

    Multiplying the first equation of Problem $ (3.1) $ by $ \omega_{t} $ and integrating over $ \Omega $, we have

    $ 12ddt[||ωt||22+(1t0g(τ)dτ)||ω||22+||ωt||22+(gω)]+12g(t)||ω||22=Ω(|ut|m(x)2ut|zt|m(x)2zt)(utzt)dx+12(gω),
    $

    from the inequality

    $ (|a|m(x)2a|b|m(x)2b)(ab)0,
    $
    (3.24)

    for all $ a, b\in R^{n} $ and a.e. $ x\in\Omega $, we obtain

    $ ||\omega_{t}||_{2}^{2}+l||\triangle \omega||_{2}^{2}+||\triangle \omega_{t}||_{2}^{2} = 0, $

    which implies that $ \omega = 0 $. This completes the proof.

    Theorem 3.1. Assume that $ (1.4) $ and $ (1.6) $ hold, let the initial date $ (u_{0}, u_{1})\in H_{0}^{2}(\Omega)\times H_{0}^{2}(\Omega) $, and

    $ 2\leq p^{-}\leq p(x)\leq p^{+}\leq\frac{2(n-3)}{n-4}, $

    then there exists a unique local solution of Problem $ (1.1) $.

    Proof. For any $ T > 0 $, consider $ M_{T} = \{u\in\mathscr H:u(0) = u_{0}, u_{t}(0) = u_{1}, ||u||_{\mathscr H}\leq M\} $. Lemma $ 3.1 $ implies that for $ \forall v\in M_{T} $, there exists $ u = S(v) $ such that $ u $ is the unique solution to Problem $ 3.1 $. Next, we prove that for a suitable $ T > 0 $, $ S $ is a contractive map satisfying $ S(M_{T})\subset M_{T} $.

    Multiplying the first equation of the Problem $ (3.1) $ by $ u_{t} $ and integrating it over $ (0, t) $, we obtain

    $ ||ut||22+||ut||22+(gu)+l||Δu||22||u1||22+||u1||22+||Δu0||22+2t0Ω|v|p(x)2vutdxdτ,
    $
    (3.25)

    using Hölder's inequality and Young's inequality, we have

    $ |Ω|v|p(x)2vutdx|γ||ut||22+14γΩ|v|2p(x)2dxγ||ut||22+14γ[Ω|v|2p2dx+Ω|v|2p+2dx]γ||ut||22+C4γ[||v||2p22+||v||2p+22],
    $

    thus, $ (3.25) $ becomes

    $ ||ut||22+||ut||22+l||Δu||22λ0+2t0Ω|v|p(x)2vutdxdτλ0+2γTsup(0,T)||ut||22+TC2γsup(0,T)[||v||2p22+||v||2p+22],
    $

    hence, we have

    $ sup(0,T)||ut||22+sup(0,T)||ut||22+lsup(0,T)||Δu||22λ0+2γTsup(0,T)||ut||22+TC2γsup(0,T)[||v||2p2H+||v||2p+2H],
    $

    where $ \lambda_{0} = ||u_{1}||_{2}^{2}+||\triangle u_{1}||_{2}^{2}+||\Delta u_{0}||_{2}^{2} $, choosing $ \gamma = \frac{1}{2T} $ such that

    $ ||u||2Hλ0+T2Csup(0,T)[||v||2p2H+||v||2p+2H].
    $

    For any $ v\in M_{T} $, by choosing $ M $ large enough so that

    $ ||u||2Hλ0+2T2CM2(p+1)M2,
    $

    and $ T > 0 $, sufficiently small so that

    $ T\leq\sqrt{\frac{M^{2}-\lambda_{0}}{2CM^{2(p^{+}-1)}}}, $

    we obtain $ ||u||_{\mathscr H}\leq M $, which shows that $ S(M_{T})\subset M_{T} $.

    Let $ v_{1}, v_{2}\in M_{T}, u_{1} = S(v_{1}), u_{2} = S(v_{2}), u = u_{1}-u_{2} $, then $ u $ satisfies

    $ {utt+2u+2uttt0g(tτ)2u(τ)dτ+|u1t|m(x)2u1t|u2t|m(x)2u2t=|v1|p(x)2v1|v2|p(x)2v2,(x,t)Ω×(0,T),u(x,t)=uν(x,t)=0,(x,t)Ω×(0,T),u(x,0)=0, ut(x,0)=0,xΩ.
    $

    Multiplying by $ u_{t} $ and integrating over $ \Omega\times(0, t) $, we obtain

    $ 12||ut||22+12(1t0g(τ)dτ)||u||22+12||ut||22+12(gu)+t0Ω[|u1t|m(x)2u1t|u2t|m(x)2u2t](u1tu2t)dxdτt0Ω(f(v1)f(v2))utdxdτ,
    $
    (3.26)

    where $ f(v) = |v|^{p(x)-2}v $. From $ (1.6) $ and $ (3.24) $, we obtain

    $ 12||ut||22+l2||u||22+12||ut||22+12(gu)t0Ω(f(v1)f(v2))utdxdτ.
    $
    (3.27)

    Now, we evaluate

    $ I = \int_{\Omega}|(f(v_{1})-f(v_{2}))||u_{t}|dx = \int_{\Omega}|f'(\xi)||v||u_{t}|dx, $

    where $ v = v_{1}-v_{2} $ and $ \xi = \alpha v_{1}+(1-\alpha)v_{2} $, $ 0\leq\alpha\leq1 $. Thanks to Young's inequality and Hölder's inequality, we have

    $ Iδ2||ut||22+12δΩ|f(ξ)|2|v|2dxδ2||ut||22+(p+1)22δΩ|ξ|2(p(x)2)|v|2dxδ2||ut||22+(p+1)22δ(Ω|v|2nn2dx)n2n[Ω|ξ|n(p(x)2)dx]2nδ2||ut||22+(p+1)22δ(Ω|v|2nn2dx)n2n[(Ω|ξ|n(p+2)dx)2n+(Ω|ξ|n(p2)dx)2n]δ2||ut||22+(p+1)2C2δ||Δv||22[||ξ||2(p+2)2+||ξ||2(p2)2]δ2||ut||22+(p+1)2C2δ||Δv||22(M2(p+2)+M2(p2)).
    $
    (3.28)

    Inserting $ (3.28) $ into $ (3.27) $, choosing $ \delta $ small enough, we obtain

    $ ||u||_{\mathscr H}^{2}\leq\frac{(p^{+}-1)^{2}CT}{\delta}(M^{2(p^{-}-2)}+M^{2(p^{+}-2)})||v||_{\mathscr H}^{2}, $

    taking $ T $ small enough so that $ \frac{(p^{+}-1)^{2}CT}{\delta}(M^{2(p^{-}-2)}+M^{2(p^{+}-2)}) < 1 $, we conclude

    $ ||u||_{\mathscr H}^{2} = ||S(v_{1})-S(v_{2})||_{\mathscr H}^{2}\leq||v_{1}-v_{2}||_{\mathscr H}^{2}, $

    thus, the contraction mapping principle ensures the existence of a weak solution to Problem $ (1.1) $. This completes the proof.

    In this section, we show that the solution to Problem $ (1.1) $ blows up in finite time when the initial energy lies in positive as well as nonpositive. For this task, we define

    $ E(t)=12||ut||22+12(1t0g(τ)dτ)||u||22+12||ut||22+12(gu)Ω1p(x)|u|p(x)dx,
    $
    (4.1)

    by the definition of $ E(t) $, we also have

    $ E(t)=Ω|ut|m(x)dx+12(gu)12g(t)||u||220.
    $
    (4.2)

    Now, we set

    $ B_{1} = \max\left\{1, \frac{B}{l^{\frac{1}{2}}}\right\}, \ \ \lambda_{1} = \left(B_{1}^{2}\right)^{\frac{-2}{p^{-}-2}}, \ \ E_{1} = (\frac{1}{2}-\frac{1}{p^{-}})(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}}, $

    and

    $ H(t)=E2E(t),
    $
    (4.3)

    where the constant $ E_{2}\in(E(0), E_{1}) $ will be discussed later, and $ B $ is the best constant of the Sobolev embedding $ H_{0}^{2}(\Omega)\hookrightarrow L^{p(x)}(\Omega) $. It follows from $ (4.2) $ that

    $ H(t)=E(t)0,
    $
    (4.4)

    and $ H(t) $ is a non$ - $decreasing function.

    To prove Theorem $ 4.1 $, we need the following two lemmas:

    Lemma 4.1. Suppose that $ (1.6) $ holds and the exponents $ m(x) $ and $ p(x) $ satisfy condition $ (1.4) $ and $ (1.5) $. Assume further that

    $ E(0) < E_{1} \ \ and\ \ \lambda_{1} < \lambda(0) = B_{1}^{2}l||\triangle u_{0}||_{2}^{2}, $

    then there exists a constant $ \lambda_{2} > \lambda_{1} $ such that

    $ B21l||u||22λ2,  t0.
    $
    (4.5)

    Proof. Using $ (1.6) $, $ (4.1) $, Lemma $ 2.4 $, and the embedding $ H_{0}^{2}(\Omega)\hookrightarrow L^{p(x)}(\Omega) $, we find that

    $ E(t)12(1t0g(τ)dτ)||u||22Ω1p(x)|u|p(x)dxl2||u||221pΩ|u|p(x)dxl2||u||221pmax{||u||pp(x),||u||p+p(x)}l2||u||221pmax{Bp||u||p2,Bp+||u||p+2}l2||u||221pmax{Bp1lp2||u||p2,Bp+1lp+2||u||p+2}12B21λ1pmax{λp2,λp+2}:=G(λ),
    $
    (4.6)

    where $ \lambda: = \lambda(t) = B_{1}^{2}l||\triangle u||_{2}^{2}. $ Analyzing directly the properties of $ G(\lambda) $, we deduce that $ G(\lambda) $ satisfies the following properties:

    $ G'(\lambda) = {12B21p+2pλp+22<0,  λ>1,12B2112λp22, 0<λ<1, 
    $
    $ G'_{+}(1) = \frac{1}{2B_{1}^{2}}-\frac{p^{+}}{2p^{-}} < 0, \ \ G'_{-}(1) = \frac{1}{2B_{1}^{2}}-\frac{1}{2} < 0, $
    $ G'(\lambda_{1}) = 0, \ \ \ \ 0 < \lambda_{1} < 1. $

    It is easily verified that $ G(\lambda) $ is strictly increasing for $ 0 < \lambda < \lambda_{1} $, strictly decreasing for $ \lambda_{1} < \lambda $, $ G(\lambda)\rightarrow -\infty $ as $ \lambda\rightarrow +\infty $, and $ G(\lambda_{1}) = E_{1} $. Since $ E(0) < E_{1} $, there exists a $ \lambda_{2} > \lambda_{1} $ such that $ G(\lambda_{2}) = E(0) $. By $ (4.6) $, we see that $ G(\lambda(0))\leq E(0) = G(\lambda_{2}) $, which implies $ \lambda(0)\geq\lambda_{2} $ since the condition $ \lambda(0) > \lambda_{1}. $ To prove $ (4.5) $, we suppose by contradiction that for some $ t_{0} > 0 $, $ \lambda_{t_{0}} = B_{1}^{2}l||\triangle u(t_{0})||_{2}^{2} < \lambda_{2} $. The continuity of $ B_{1}^{2}l||\triangle u||_{2}^{2} $ illustrates that we could choose $ t_{0} $ such that $ \lambda_{1} < \lambda_{t_{0}} < \lambda_{2} $, then we have $ E(0) = G(\lambda_{2}) < G(\lambda_{t_{0}})\leq E(t_{0}) $. This is a contradiction. The proof is completed.

    Lemma 4.2. Let the assumption in Lemma $ 4.1 $ be satisfied. For $ t\in[0, T) $, we have

    $ 0<H(0)H(t)1pρp(x)(u).
    $

    Proof. $ (4.4) $ indicates that $ H(t) $ is nondecreasing with respect to $ t $, thus

    $ H(t)\geq H(0) = E_{2}-E(0) > 0, \ \ \forall t\in[0, T). $

    It follows from $ (1.6) $, $ (4.1) $, and Lemma $ 4.1 $ that

    $ H(t)=E2E(t)=E212||ut||2212(1t0g(τ)dτ)||u||2212(gu)12||ut||22+Ω1p(x)|u|p(x)dxE1l2||u||22+Ω1p(x)|u|p(x)dxE112B21λ2+Ω1p(x)|u|p(x)dxE112B21λ1+Ω1p(x)|u|p(x)dxΩ1p(x)|u|p(x)dx1pρp(x)(u).
    $

    The proof is completed.

    Our blow-up result reads as follows:

    Theorem 4.3. Suppose that

    $ 2\leq m^{-}\leq m(x)\leq m^{+} < p^{-}\leq p(x)\leq p^{+}\leq\frac{2(n-3)}{n-4}, $

    and

    $ 1l=0g(τ)dτ<p21p21+12p,
    $
    (4.7)

    hold, if the following conditions

    $ E(0) < \frac{1}{2}(\frac{1}{2}-\frac{1}{p^{-}})\left(1-\frac{1}{p^{-}(p^{-}-2)}\frac{1-l}{l}\right)(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}} \ \ and\ \ \lambda_{1} < \lambda(0) = B_{1}^{2}l||\triangle u_{0}||_{2}^{2}, $

    are satisfied, then there exists $ T^{\ast} < +\infty $ such that

    $ limtT(||ut||22+||ut||22+||u||22+||u||p+p+)=+.
    $
    (4.8)

    Proof. Assume by contradiction that $ (4.8) $ does not hold true, then for $ \forall T^{\ast} < +\infty $ and all $ t\in[0, T^{\ast}] $, we get

    $ ||ut||22+||ut||22+||u||22+||u||p+p+C,
    $
    (4.9)

    where $ C_{\ast} $ is a positive constant.

    Now, we define $ L(t) $ as follows:

    $ L(t)=H1α(t)+ϵΩutudx+ϵΩutudx,
    $
    (4.10)

    where $ \varepsilon > 0 $, small enough to be chosen later, and

    $ 0αmin{pm+p(m+1),p22p}.
    $

    The remaining proof will be divided into two steps.

    $ \textbf{Step} $ $ \textbf{1:} $ $ \textbf{Estimate} $ $ \textbf{for} $ $ \textbf{L'(t)}. $ By taking the derivative of $ (4.10) $ and using $ (1.1) $, we obtain

    $ L(t)=(1α)Hα(t)[Ω|ut|m(x)dx12(gu)+12g(t)||u||22]+ϵ||ut||22+ϵΩuttudx+ϵ||ut||22ϵ||u||22+ϵΩt0g(tτ)u(τ)dτudxϵΩ|ut|m(x)2utudx+ϵΩ|u|p(x)dxϵΩuttudx(1α)Hα(t)Ω|ut|m(x)dx+ϵ||ut||22ϵ||u||22+ϵΩt0g(tτ)u(τ)dτudxϵΩ|ut|m(x)2utudx+ϵΩ|u|p(x)dx+ϵ||Δut||22,
    $

    applying Hölder's inequality and Young's inequality, we have

    $ ϵΩt0g(tτ)Δu(τ)Δu(t)dτdx=ϵΩt0g(tτ)Δu(t)(Δu(τ)Δu(t))dτdx+ϵt0g(tτ)dτ||Δu||22ϵt0g(tτ)||Δu(τ)Δu(t)||2||Δu(t)||2dτ+ϵt0g(tτ)dτ||Δu||22ϵp(1ε1)2(gu)+ϵ(112p(1ε1))t0g(τ)dτ||Δu||22,
    $

    where $ 0 < \varepsilon_{1} < \frac{p^{-}-2}{p^{-}} $, then

    $ L(t)(1α)Hα(t)Ω|ut|m(x)dx+ϵ||ut||22ϵ||Δu||22+ϵ||Δut||22ϵΩ|ut|m(x)2utudx+ϵΩ|u|p(x)dxϵp(1ε1)2(gu)+ϵ(112p(1ε1))t0g(τ)dτ||Δu||22,
    $

    rewriting $ (4.7) $ to $ (\frac{p^{-}}{2}-1)l-\frac{1}{2p^{-}}(1-l) > 0 $, using $ (4.1) $ and $ (4.3) $ to substitute for $ (g\diamond \bigtriangleup u) $, choosing $ \varepsilon_{1} > 0 $ sufficiently small, we obtain

    $ L(t)(1α)Hα(t)Ω|ut|m(x)dx+ϵp(1ε1)H(t)+(ϵ+ϵp(1ε1)2)(||ut||22+||ut||22)+ϵ{(p(1ε1)21)(1t0g(τ)dτ)12p(1ε1)t0g(τ)dτ}||u||22ϵp(1ε1)E2ϵΩ|ut|m(x)2utudx+ϵε1Ω|u|p(x)dx(1α)Hα(t)Ω|ut|m(x)dx+ϵp(1ε1)H(t)+(ϵ+ϵp(1ε1)2)(||ut||22+||ut||22)+ϵ{(p(1ε1)21)l212p(1ε1)1l2}lλ2B21ϵp(1ε1)E2ϵΩ|ut|m(x)2utudx+ϵ{(p(1ε1)21)l212p(1ε1)1l2}||u||22+ϵε1Ω|u|p(x)dx.(1α)Hα(t)Ω|ut|m(x)dx+ϵp(1ε1)H(t)+(ϵ+ϵp(1ε1)2)(||ut||22+||ut||22)+ϵ{(p(1ε1)21)l212p(1ε1)1l2}l(B21)pp2ϵp(1ε1)E2ϵΩ|ut|m(x)2utudx+ϵ{(p(1ε1)21)l212p(1ε1)1l2}||u||22+ϵε1Ω|u|p(x)dx.
    $
    (4.11)

    $ \textbf{Step} $ $ \textbf{1.1:} $ $ \textbf{Estimate} $ $ \textbf{for} $ $ \epsilon\frac{\left\{\left(\frac{p^{-}(1-\varepsilon_{1})}{2}-1\right)\frac{l}{2}-\frac{1}{2p^{-}(1-\varepsilon_{1})}\frac{1-l}{2}\right\}}{l}(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}}-\epsilon p^{-}(1-\varepsilon_{1})E_{2}. $ It follows from the condition in Theorem $ 3.1 $ that

    $ E(0) < \frac{1}{2}(\frac{1}{2}-\frac{1}{p^{-}})\left(1-\frac{1-l}{p^{-}(p^{-}-2)l}\right)(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}} = \frac{(\frac{p^{-}}{2}-1)\frac{l}{2}-\frac{1}{2p^{-}}\frac{(1-l)}{2}}{lp^{-}}(B_{1}^{2})^{\frac{-p^{-}}{p^{-}-2}} < E_{1}, $

    here, we can take $ \varepsilon_{1} > 0 $ sufficiently small and choose $ E_{2}\in(E(0), E_{1}) $ sufficiently close to $ E(0) $ such that

    $ ϵ(p(1ε1)21)l212p(1ε1)(1l)2l(B21)pp2ϵ(1ε1)pE2ϵ(p(1ε1)21)l212p(1ε1)(1l)2l(B21)pp2ϵ(1ε1)p(p21)l212p(1l)2lp(B21)pp20.
    $
    (4.12)

    Therefore, we obtain by combining $ (4.11) $ and $ (4.12) $,

    $ L(t)(1α)Hα(t)Ω|ut|m(x)dx+ϵp(1ε1)H(t)+(ϵ+ϵp(1ε1)2)(||ut||22+||ut||22)+ϵε1Ω|u|p(x)dx+ϵ{(p(1ε1)21)l212p(1ε1)1l2}||u||22ϵΩ|ut|m(x)2utudx.
    $
    (4.13)

    $ \textbf{Step} $ $ \textbf{1.2:} $ $ \textbf{Estimate} $ $ \textbf{for} $ $ -\epsilon \int_{\Omega}|u_{t}|^{m(x)-2}u_{t}udx. $ Applying Young's inequality with $ \varepsilon_{2} > 1 $, the embedding $ L^{p(x)}(\Omega)\hookrightarrow L^{m(x)}(\Omega) $, Lemma $ 2.4 $ and Lemma $ 4.2 $, we easily have

    $ |Ω|ut|m(x)2utudx|Ω|ut|m(x)1Hαm(x)1m(x)(t)Hαm(x)1m(x)(t)|u|dxε2Hα(t)Ω|ut|m(x)dx+1εm12Ω|u|m(x)Hα(m(x)1)(t)dxε2Hα(t)Ω|ut|m(x)dx+2Cα(mm+)1εm12Hα(m+1)(t)Ω|u|m(x)dxε2Hα(t)Ω|ut|m(x)dx+C2εm12Hα(m+1)(t)max{||u||m+p(x),||u||mp(x)},
    $
    (4.14)

    where $ C_{1} = \min\left\{H(0), 1\right\} $, $ C_{2} = 2(1+|\Omega|)^{m^{+}}C_{1}^{\alpha(m^{-}-m^{+})} $. Next, we have

    $ ||u||m+p(x)max{(Ω|u|p(x)dx)m+p+,(Ω|u|p(x)dx)m+p}max{[pH(t)]m+p+m+p,1}(Ω|u|p(x)dx)m+p,
    $

    and

    $ ||u||mp(x)max{[pH(t)]mp+m+p,[pH(t)]mm+p}(Ω|u|p(x)dx)m+p,
    $

    which illustrate

    $ max{||u||m+p(x),||u||mp(x)}C3(Ω|u|p(x)dx)m+p,
    $

    where $ C_{3} = 2\min\left\{p^{-}H(0), 1\right\}^{\frac{m^{-}}{p^{+}}-\frac{m^{+}}{p^{-}}} $. Recalling $ 0 < \alpha\leq\frac{p^{-}-m^{+}}{p^{-}(m^{+}-1)} $ and Lemma $ 4.2 $, apparently,

    $ Hα(m+1)(t)max{||u||m+p(x),||u||mp(x)}C3Hα(m+1)(t)(Ω|u|p(x)dx)m+pC3Hα(m+1)+m+p1(t)Hα(m+1)+m+p1(0)H1m+p(t)Hα(m+1)+m+p1(0)(Ω|u|p(x)dx)m+pC3(1p)1m+p(Ω|u|p(x)dx)1m+pHα(m+1)+m+p1(0)(Ω|u|p(x)dx)m+pC3(1p)1m+pCα(m+1)+m+p11Ω|u|p(x)dx,
    $
    (4.15)

    it follows from $ (4.13) $, $ (4.14) $, and $ (4.15) $ that

    $ L(t)(1αϵε2)Hα(t)Ω|ut|m(x)dx+(ϵ+ϵp(1ε1)2)(||ut||22+||ut||22)+ϵ(1ε1)pH(t)+ϵ(ε1Cα(m+1)+m+p11C2C3(1p)1m+pεm12)Ω|u|p(x)dx+ϵ{(p(1ε1)21)l212p(1ε1)1l2}||u||22,
    $

    let us fix the constant $ \varepsilon_{2} $ so that

    $ \varepsilon_{1} > \frac{C_{1}^{\alpha(m^{+}-1)+\frac{m^{+}}{p^{-}}-1}C_{2}C_{3}(\frac{1}{p^{-}})^{1-\frac{m^{+}}{p^{-}}}}{\varepsilon_{2}^{m^{-}-1}}, $

    and then choose $ \epsilon $ so small that $ 1-\alpha > \epsilon\varepsilon_{1} $. Therefore, we obtain

    $ L(t)M1(H(t)+||u||22+||ut||22+||ut||22+Ω|u|p(x)dx),
    $
    (4.16)

    where

    $ M1=ϵmin{(1+p(1ε1)2),(1ε1)p,ε1Cα(m+1)+m+p11C2C3(1p)1m+pεm12,,(p(1ε1)21)l212p(1ε1)1l2}.
    $

    Inequalities $ (4.16) $ and Lemma $ 4.2 $ imply $ L(t)\geq L(0). $ Therefore, for a sufficiently small $ \epsilon $, we have

    $ L(0) = H^{1-\alpha}(0)+\epsilon\int_{\Omega}u_{1}u_{0}dx+\epsilon\int_{\Omega}\triangle u_{1}\triangle u_{0}dx > 0. $

    $ \textbf{Step} $ $ \textbf{2:} $ $ \textbf{A} $ $ \textbf{differential} $ $ \textbf{inequality} $ $ \textbf{for} $ $ \textbf{L(t)}. $ Applying Hölder's inequality, Young's inequality and the embedding $ L^{p(x)}(\Omega)\hookrightarrow L^{2}(\Omega) $, we easily obtain

    $ |Ωutudx|11α(ut2u2)11α(1+|Ω|)11α||ut||11α2||u||11αp(x)(1+|Ω|)11αμ||ut||11αμ2+(1+|Ω|)11αν||u||11ανp(x),
    $
    (4.17)

    where $ \frac{1}{\mu}+\frac{1}{\nu} = 1.\ $Choosing $ \mu = 2(1-\alpha) > 1 $, then $ \nu = \frac{2(1-\alpha)}{2(1-\alpha)-1}, \ $further, $ (4.17) $ can be rewritten as

    $ |Ωutudx|11α(1+|Ω|)11αμ||ut||22+(1+|Ω|)11αν||u||22(1α)1p(x),
    $
    (4.18)

    recalling $ 0 < \alpha < \frac{p^{-}-2}{2p^{-}} $, we obtain

    $ ||u||22(1α)1p(x)max{(Ω|u|p(x)dx)2p[2(1α)1],(Ω|u|p(x)dx)2p+[2(1α)1]}{[pH(t)]2p[2(1α)1]p[2(1α)1],[pH(t)]2p+[2(1α)1]p+[2(1α)1]}Ω|u|p(x)dxC4Ω|u|p(x)dx,
    $
    (4.19)

    with $ C_{4} = \min\{p^{-}H(0), 1\}^{\frac{2-p^{+}[2(1-\alpha)-1]}{p^{+}[2(1-\alpha)-1]}} $. Inserting $ (4.19) $ into $ (4.18) $, we obtain

    $ |Ωutudx|11α(1+|Ω|)11αμ||ut||22+(1+|Ω|)11ανC4Ω|u|p(x)dx.
    $
    (4.20)

    We now estimate

    $ |Ωutudx|11α||ut||11α2||Δu||11α2C11αC11αH(0)H(t),
    $
    (4.21)

    therefore, combining $ (4.20) $ and $ (4.21) $, we obtain

    $ L11α(t)=(H1α(t)+ϵΩutudx+ϵΩutudx)11αM2(H(t)+||ut||22+||ut||22+||u||22+Ω|u|p(x)dx),
    $
    (4.22)

    where

    $ M2=max{211α(211α+ϵ11αC11αH(0)), 221αϵ11α(1+|Ω|)11αμ, 221αϵ11α(1+|Ω|)11ανC4}.
    $

    Combining $ (4.16) $ and $ (4.22) $, we arrive at

    $ L(t)M1M2L11α(t),t0.
    $
    (4.23)

    A simple integration of $ (4.23) $ over $ (0, t) $ yields

    $ Lα1α(t)1Lαα1(0)M1M2α1αt,
    $

    this shows that $ L(t) $ blows up in finite time

    $ TM2M11ααLαα1(0),
    $

    furthermore, one gets from $ (4.22) $ that

    $ \lim\limits_{t\rightarrow T^{\ast-}}\left(H(t)+||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2}+||\triangle u||_{2}^{2}+\int_{\Omega}|u|^{p(x)}dx\right) = +\infty, $

    it easily follows that

    $ \int_{\Omega}|u|^{p(x)}dx\leq \int_{\{|u|\geq1\}}|u|^{p^{+}}dx+\int_{\{|u| < 1\}}|u|^{p^{-}}dx\leq||u||_{p^{+}}^{p^{+}}+|\Omega|, $

    and using Lemma $ 4.2 $, we obtain

    $ \lim\limits_{t\rightarrow T^{\ast-}}\left(||u_{t}||_{2}^{2}+||\triangle u_{t}||_{2}^{2}+||\triangle u||_{2}^{2}+||u||_{p^{+}}^{p^{+}}\right) = +\infty, $

    this leads to a contradiction with $ (4.9) $. Thus, the solution to Problem $ (1.1) $ blows up in finite time.

    Ying Chu: Methodology, Wring-original draft, Writing-review editing; Bo Wen and Libo Cheng: Methodology, Writing-original draft.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors express their heartfelt thanks to the editors and referees who have provided some important suggestions. This work was supported by Science and Technology Development Plan Project of Jilin Province, China (20240101307JC).

    The authors declare there is no conflict of interest.

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