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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+△2utt−∫t0g(t−τ)△2u(τ)dτ+|ut|m(x)−2ut=|u|p(x)−2u,(x,t)∈Ω×(0,T),u(x,t)=∂u∂ν(x,t)=0,(x,t)∈∂Ω×(0,T),u(x,0)=u0(x), ut(x,0)=u1(x),x∈Ω, $
|
(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,y∈Ω,|x−y|<1,|m(x)−m(y)|+|p(x)−p(y)|≤ω(|x−y|), $
|
(1.2) |
where
$ limτ→0+supω(τ)ln1τ=C<∞. $
|
(1.3) |
In addition to this condition, the exponents satisfy the following:
$ 2≤m−:=essinfx∈Ωm(x)≤m(x)≤m+:=esssupx∈Ωm(x)<2(n−2)n−4, $
|
(1.4) |
$ 2≤p−:=essinfx∈Ωp(x)≤p(x)≤p+:=esssupx∈Ωp(x)<2(n−2)n−4, $
|
(1.5) |
$ g: R^{+} \, \rightarrow R^{+} $ is a $ C^{1} $ function satisfying
$ g(0)>0, g′(τ)≤0,1−∫∞0g(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
$ utt−△u+∫t0g(t−τ)△u(τ)dτ+|ut|m−2ut=|u|p−2u, $
|
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|ρutt−△u+∫t0g(t−τ)△u(τ)dτ+|ut|m−2ut=|u|p−2u, $
|
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|ρutt−△u−△utt+∫t0g(t−τ)△u(τ)dτ−γ△ut=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:
$ utt−△u+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
$ utt−△u+∫t0g(t−s)△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
$ utt−△u+∫t0g(t−τ)△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+△2u−△u+|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+△2u−M(||∇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)n−kp(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+△2utt−∫t0g(t−τ)△2u(τ)dτ+|ut|m(x)−2ut=|v|p(x)−2v,(x,t)∈Ω×(0,T),u(x,t)=∂u∂ν(x,t)=0,(x,t)∈∂Ω×(0,T),u(x,0)=u0(x), ut(x,0)=u1(x),x∈Ω. $
|
(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(t−τ)(△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|k∑j=1ckjt(t)ωj(x)|m(x)−2k∑j=1ckjt(t)ωj(x)ωi(x)dx, $
|
and
$ gi(cki(t))=λ2i∫t0g(t−τ)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)dx−∫t0g(t−τ)∫Ω△uk(τ)△uktdxdτ=∫Ω|v|p(x)−2vuktdx. $
|
(3.4) |
By simple calculation, we have
$ −∫t0g(t−τ)∫ΩΔuk(τ)Δuktdxdτ=12ddt(g⋄△uk)−12(g′⋄△uk)−12ddt∫t0g(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(g⋄△uk)+12(1−∫t0g(τ)dτ)||Δuk||22]=12(g′⋄△uk)−12g(t)||△uk||22+∫Ω|v|p(x)−2vuktdx−∫Ω|ukt|m(x)dx≤∫Ω|v|p(x)−2vuktdx≤‖|v|p(x)−2v‖2||ukt||2≤η2∫Ω|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)dx≤max{||v||2(p−−1)2(p(x)−1),||v||2(p+−1)2(p(x)−1)}≤Cmax{||△v||2(p−−1)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(g⋄△uk)+12(1−∫t0g(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+(g⋄△uk)+l||Δuk||22≤CT, $
|
(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
$ uk∗⇀u weakly star in L∞(0,T;H20(Ω)),ukt∗⇀ut weakly star in L∞(0,T;H20(Ω)),uk⇀u weakly in L2(0,T;H20(Ω)),ukt⇀ut 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)=−∫Ω△uk△ukttdx+∫t0g(t−τ)∫ΩΔuk(τ)Δukttdxdτ+∫Ω|v|p(x)−2vukttdx. $
|
(3.10) |
Note that we have the estimates for $ \varepsilon > 0 $
$ |∫Ω△uk△ukttdx|≤ε||△uktt||22+14ε||△uk||22, $
|
(3.11) |
$ ∫Ω|v|p(x)−2vukttdx≤‖|v|p(x)−2v‖2‖uktt‖2≤ε||uktt||22+14ε∫Ω|v|2(p(x)−1)dx, $
|
(3.12) |
and
$ |∫t0g(t−τ)∫Ω△uk(τ)△ukttdxdτ|≤14ε∫Ω(∫t0g(t−τ)△uk(τ)dτ)2dx+ε||△uktt||22≤ε||△uktt||22+14ε∫t0g(s)ds∫t0g(t−τ)∫Ω|△uk(τ)|2dxdτ≤ε||△uktt||22+(1−l)g(0)4ε∫t0||△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)−2vukttdx≤εC||△uktt||22+C4ε. $
|
(3.14) |
Taking into account $ (3.10)-(3.14) $, we obtain
$ ||uktt||22+(1−2ε−Cε)||△uktt||22+ddt(∫Ω1m(x)|ukt|m(x)dx)≤14ε||△uk||22+(1−l)g(0)4ε∫t0||△uk(τ)||22dτ+C4ε, $
|
(3.15) |
integrating $ (3.15) $ over $ (0, t) $, we obtain
$ ∫t0||uktt||22dτ+(1−2ε−Cε)∫t0||△uktt||22dτ+∫Ω1m(x)|ukt|m(x)dx≤C4ε∫t0(||△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||22dτ≤CT, $
|
(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
$ uktt⇀utt 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;H−2(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
$ ukt→ut 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)−2ukt‖22=∫Ω|ukt|2(m(x)−1)dx≤max{||△ukt||2(m−−1)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+△2ωtt−∫t0g(t−τ)△2ω(τ)dτ+|ut|m(x)−2ut−|zt|m(x)−2zt=0,(x,t)∈Ω×(0,T),ω(x,t)=∂ω∂ν(x,t)=0,(x,t)∈∂Ω×(0,T),ω(x,0)=0, ωt(x,0)=0,x∈Ω. $
|
Multiplying the first equation of Problem $ (3.1) $ by $ \omega_{t} $ and integrating over $ \Omega $, we have
$ 12ddt[||ωt||22+(1−∫t0g(τ)dτ)||△ω||22+||△ωt||22+(g⋄△ω)]+12g(t)||△ω||22=−∫Ω(|ut|m(x)−2ut−|zt|m(x)−2zt)(ut−zt)dx+12(g′⋄△ω), $
|
from the inequality
$ (|a|m(x)−2a−|b|m(x)−2b)(a−b)≥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+(g⋄△u)+l||Δu||22≤||u1||22+||△u1||22+||Δu0||22+2∫t0∫Ω|v|p(x)−2vutdxdτ, $
|
(3.25) |
using Hölder's inequality and Young's inequality, we have
$ |∫Ω|v|p(x)−2vutdx|≤γ||ut||22+14γ∫Ω|v|2p(x)−2dx≤γ||ut||22+14γ[∫Ω|v|2p−−2dx+∫Ω|v|2p+−2dx]≤γ||ut||22+C4γ[||△v||2p−−22+||△v||2p+−22], $
|
thus, $ (3.25) $ becomes
$ ||ut||22+||△ut||22+l||Δu||22≤λ0+2∫t0∫Ω|v|p(x)−2vutdxdτ≤λ0+2γTsup(0,T)||ut||22+TC2γsup(0,T)[||△v||2p−−22+||△v||2p+−22], $
|
hence, we have
$ sup(0,T)||ut||22+sup(0,T)||△ut||22+lsup(0,T)||Δu||22≤λ0+2γTsup(0,T)||ut||22+TC2γsup(0,T)[||v||2p−−2H+||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||2H≤λ0+T2Csup(0,T)[||v||2p−−2H+||v||2p+−2H]. $
|
For any $ v\in M_{T} $, by choosing $ M $ large enough so that
$ ||u||2H≤λ0+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+△2utt−∫t0g(t−τ)△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)=∂u∂ν(x,t)=0,(x,t)∈∂Ω×(0,T),u(x,0)=0, ut(x,0)=0,x∈Ω. $
|
Multiplying by $ u_{t} $ and integrating over $ \Omega\times(0, t) $, we obtain
$ 12||ut||22+12(1−∫t0g(τ)dτ)||△u||22+12||△ut||22+12(g⋄△u)+∫t0∫Ω[|u1t|m(x)−2u1t−|u2t|m(x)−2u2t](u1t−u2t)dxdτ≤∫t0∫Ω(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(g⋄△u)≤∫t0∫Ω(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
$ I≤δ2||ut||22+12δ∫Ω|f′(ξ)|2|v|2dx≤δ2||ut||22+(p+−1)22δ∫Ω|ξ|2(p(x)−2)|v|2dx≤δ2||ut||22+(p+−1)22δ(∫Ω|v|2nn−2dx)n−2n[∫Ω|ξ|n(p(x)−2)dx]2n≤δ2||ut||22+(p+−1)22δ(∫Ω|v|2nn−2dx)n−2n[(∫Ω|ξ|n(p+−2)dx)2n+(∫Ω|ξ|n(p−−2)dx)2n]≤δ2||ut||22+(p+−1)2C2δ||Δv||22[||△ξ||2(p+−2)2+||△ξ||2(p−−2)2]≤δ2||ut||22+(p+−1)2C2δ||Δv||22(M2(p+−2)+M2(p−−2)). $
|
(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(1−∫t0g(τ)dτ)||△u||22+12||△ut||22+12(g⋄△u)−∫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(g′⋄△u)−12g(t)||△u||22≤0. $
|
(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)=E2−E(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||22≥λ2, t≥0. $
|
(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(1−∫t0g(τ)dτ)||△u||22−∫Ω1p(x)|u|p(x)dx≥l2||△u||22−1p−∫Ω|u|p(x)dx≥l2||△u||22−1p−max{||u||p−p(x),||u||p+p(x)}≥l2||△u||22−1p−max{Bp−||△u||p−2,Bp+||△u||p+2}≥l2||△u||22−1p−max{Bp−1lp−2||△u||p−2,Bp+1lp+2||△u||p+2}≥12B21λ−1p−max{λp−2,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) = {12B21−p+2p−λp+−22<0, λ>1,12B21−12λp−−22, 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)≤1p−ρp(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)=E2−E(t)=E2−12||ut||22−12(1−∫t0g(τ)dτ)||△u||22−12(g⋄△u)−12||△ut||22+∫Ω1p(x)|u|p(x)dx≤E1−l2||△u||22+∫Ω1p(x)|u|p(x)dx≤E1−12B21λ2+∫Ω1p(x)|u|p(x)dx≤E1−12B21λ1+∫Ω1p(x)|u|p(x)dx≤∫Ω1p(x)|u|p(x)dx≤1p−ρp(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
$ 1−l=∫∞0g(τ)dτ<p−2−1p−2−1+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
$ limt→T∗−(||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)=H1−α(t)+ϵ∫Ωutudx+ϵ∫Ω△ut△udx, $
|
(4.10) |
where $ \varepsilon > 0 $, small enough to be chosen later, and
$ 0≤α≤min{p−−m+p−(m+−1),p−−22p−}. $
|
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)=(1−α)H−α(t)[∫Ω|ut|m(x)dx−12(g′⋄△u)+12g(t)||△u||22]+ϵ||ut||22+ϵ∫Ω△utt△udx+ϵ||△ut||22−ϵ||△u||22+ϵ∫Ω∫t0g(t−τ)△u(τ)dτ△udx−ϵ∫Ω|ut|m(x)−2utudx+ϵ∫Ω|u|p(x)dx−ϵ∫Ω△utt△udx≥(1−α)H−α(t)∫Ω|ut|m(x)dx+ϵ||ut||22−ϵ||△u||22+ϵ∫Ω∫t0g(t−τ)△u(τ)dτ△udx−ϵ∫Ω|ut|m(x)−2utudx+ϵ∫Ω|u|p(x)dx+ϵ||Δut||22, $
|
applying Hölder's inequality and Young's inequality, we have
$ ϵ∫Ω∫t0g(t−τ)Δu(τ)Δu(t)dτdx=ϵ∫Ω∫t0g(t−τ)Δu(t)(Δu(τ)−Δu(t))dτdx+ϵ∫t0g(t−τ)dτ||Δu||22≥−ϵ∫t0g(t−τ)||Δu(τ)−Δu(t)||2||Δu(t)||2dτ+ϵ∫t0g(t−τ)dτ||Δu||22≥−ϵp−(1−ε1)2(g⋄△u)+ϵ(1−12p−(1−ε1))∫t0g(τ)dτ||Δu||22, $
|
where $ 0 < \varepsilon_{1} < \frac{p^{-}-2}{p^{-}} $, then
$ L′(t)≥(1−α)H−α(t)∫Ω|ut|m(x)dx+ϵ||ut||22−ϵ||Δu||22+ϵ||Δut||22−ϵ∫Ω|ut|m(x)−2utudx+ϵ∫Ω|u|p(x)dx−ϵp−(1−ε1)2(g⋄△u)+ϵ(1−12p−(1−ε1))∫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)≥(1−α)H−α(t)∫Ω|ut|m(x)dx+ϵp−(1−ε1)H(t)+(ϵ+ϵp−(1−ε1)2)(||ut||22+||△ut||22)+ϵ{(p−(1−ε1)2−1)(1−∫t0g(τ)dτ)−12p−(1−ε1)∫t0g(τ)dτ}||△u||22−ϵp−(1−ε1)E2−ϵ∫Ω|ut|m(x)−2utudx+ϵε1∫Ω|u|p(x)dx≥(1−α)H−α(t)∫Ω|ut|m(x)dx+ϵp−(1−ε1)H(t)+(ϵ+ϵp−(1−ε1)2)(||ut||22+||△ut||22)+ϵ{(p−(1−ε1)2−1)l2−12p−(1−ε1)1−l2}lλ2B21−ϵp−(1−ε1)E2−ϵ∫Ω|ut|m(x)−2utudx+ϵ{(p−(1−ε1)2−1)l2−12p−(1−ε1)1−l2}||△u||22+ϵε1∫Ω|u|p(x)dx.≥(1−α)H−α(t)∫Ω|ut|m(x)dx+ϵp−(1−ε1)H(t)+(ϵ+ϵp−(1−ε1)2)(||ut||22+||△ut||22)+ϵ{(p−(1−ε1)2−1)l2−12p−(1−ε1)1−l2}l(B21)−p−p−−2−ϵp−(1−ε1)E2−ϵ∫Ω|ut|m(x)−2utudx+ϵ{(p−(1−ε1)2−1)l2−12p−(1−ε1)1−l2}||△u||22+ϵε1∫Ω|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−(1−ε1)2−1)l2−12p−(1−ε1)(1−l)2l(B21)−p−p−−2−ϵ(1−ε1)p−E2≥ϵ(p−(1−ε1)2−1)l2−12p−(1−ε1)(1−l)2l(B21)−p−p−−2−ϵ(1−ε1)p−(p−2−1)l2−12p−(1−l)2lp−(B21)−p−p−−2≥0. $
|
(4.12) |
Therefore, we obtain by combining $ (4.11) $ and $ (4.12) $,
$ L′(t)≥(1−α)H−α(t)∫Ω|ut|m(x)dx+ϵp−(1−ε1)H(t)+(ϵ+ϵp−(1−ε1)2)(||ut||22+||△ut||22)+ϵε1∫Ω|u|p(x)dx+ϵ{(p−(1−ε1)2−1)l2−12p−(1−ε1)1−l2}||△u||22−ϵ∫Ω|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)−1H−αm(x)−1m(x)(t)Hαm(x)−1m(x)(t)|u|dx≤ε2H−α(t)∫Ω|ut|m(x)dx+1εm−−12∫Ω|u|m(x)Hα(m(x)−1)(t)dx≤ε2H−α(t)∫Ω|ut|m(x)dx+2Cα(m−−m+)1εm−−12Hα(m+−1)(t)∫Ω|u|m(x)dx≤ε2H−α(t)∫Ω|ut|m(x)dx+C2εm−−12Hα(m+−1)(t)max{||u||m+p(x),||u||m−p(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{[p−H(t)]m+p+−m+p−,1}(∫Ω|u|p(x)dx)m+p−, $
|
and
$ ||u||m−p(x)≤max{[p−H(t)]m−p+−m+p−,[p−H(t)]m−−m+p−}(∫Ω|u|p(x)dx)m+p−, $
|
which illustrate
$ max{||u||m+p(x),||u||m−p(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||m−p(x)}≤C3Hα(m+−1)(t)(∫Ω|u|p(x)dx)m+p−≤C3Hα(m+−1)+m+p−−1(t)Hα(m+−1)+m+p−−1(0)H1−m+p−(t)Hα(m+−1)+m+p−−1(0)(∫Ω|u|p(x)dx)m+p−≤C3(1p−)1−m+p−(∫Ω|u|p(x)dx)1−m+p−Hα(m+−1)+m+p−−1(0)(∫Ω|u|p(x)dx)m+p−≤C3(1p−)1−m+p−Cα(m+−1)+m+p−−11∫Ω|u|p(x)dx, $
|
(4.15) |
it follows from $ (4.13) $, $ (4.14) $, and $ (4.15) $ that
$ L′(t)≥(1−α−ϵε2)H−α(t)∫Ω|ut|m(x)dx+(ϵ+ϵp−(1−ε1)2)(||ut||22+||△ut||22)+ϵ(1−ε1)p−H(t)+ϵ(ε1−Cα(m+−1)+m+p−−11C2C3(1p−)1−m+p−εm−−12)∫Ω|u|p(x)dx+ϵ{(p−(1−ε1)2−1)l2−12p−(1−ε1)1−l2}||△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−(1−ε1)2),(1−ε1)p−,ε1−Cα(m+−1)+m+p−−11C2C3(1p−)1−m+p−εm−−12,,(p−(1−ε1)2−1)l2−12p−(1−ε1)1−l2}. $
|
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|11−α≤(‖ut‖2‖u‖2)11−α≤(1+|Ω|)11−α||ut||11−α2||u||11−αp(x)≤(1+|Ω|)11−αμ||ut||11−αμ2+(1+|Ω|)11−αν||u||11−ανp(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|11−α≤(1+|Ω|)11−αμ||ut||22+(1+|Ω|)11−αν||u||22(1−α)−1p(x), $
|
(4.18) |
recalling $ 0 < \alpha < \frac{p^{-}-2}{2p^{-}} $, we obtain
$ ||u||22(1−α)−1p(x)≤max{(∫Ω|u|p(x)dx)2p−[2(1−α)−1],(∫Ω|u|p(x)dx)2p+[2(1−α)−1]}≤{[p−H(t)]2−p−[2(1−α)−1]p−[2(1−α)−1],[p−H(t)]2−p+[2(1−α)−1]p+[2(1−α)−1]}∫Ω|u|p(x)dx≤C4∫Ω|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|11−α≤(1+|Ω|)11−αμ||ut||22+(1+|Ω|)11−ανC4∫Ω|u|p(x)dx. $
|
(4.20) |
We now estimate
$ |∫Ω△ut△udx|11−α≤||△ut||11−α2||Δu||11−α2≤C11−α∗≤C11−α∗H(0)H(t), $
|
(4.21) |
therefore, combining $ (4.20) $ and $ (4.21) $, we obtain
$ L11−α(t)=(H1−α(t)+ϵ∫Ωutudx+ϵ∫Ω△ut△udx)11−α≤M2(H(t)+||ut||22+||△ut||22+||△u||22+∫Ω|u|p(x)dx), $
|
(4.22) |
where
$ M2=max{211−α(211−α+ϵ11−αC11−α∗H(0)), 221−αϵ11−α(1+|Ω|)11−αμ, 221−αϵ11−α(1+|Ω|)11−ανC4}. $
|
Combining $ (4.16) $ and $ (4.22) $, we arrive at
$ L′(t)≥M1M2L11−α(t),∀t≥0. $
|
(4.23) |
A simple integration of $ (4.23) $ over $ (0, t) $ yields
$ Lα1−α(t)≥1Lαα−1(0)−M1M2α1−αt, $
|
this shows that $ L(t) $ blows up in finite time
$ T∗≤M2M11−ααLαα−1(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.
[1] |
Blikslager AT, Moeser AJ, Gookin JL, et al. (2007) Restoration of barrier function in injured intestinal mucosa. Physiol Rev 87: 545. doi: 10.1152/physrev.00012.2006
![]() |
[2] | Podolsky DK (1999) V. Innate mechanisms of mucosal defense and repair: The best offense is a good defense. Am J Physiol 277: G495–G499. |
[3] |
Kunzelmann K, Mall M (2002) Electrolyte transport in the mammalian colon: Mechanisms and Implications for disease. Physiol Rev 82: 245–289. doi: 10.1152/physrev.00026.2001
![]() |
[4] |
Ferraris RP, Diamond J (1997) Regulation of intestinal sugar transport. Physiol Rev 77: 257–302. doi: 10.1152/physrev.1997.77.1.257
![]() |
[5] |
Groschwitz KR, Hogan SP (2009) Intestinal barrier function: Molecular regulation and disease pathogenesis. J Allergy Clin Immunol 124: 3–22. doi: 10.1016/j.jaci.2009.05.038
![]() |
[6] |
Bischoff S, Barbara G, Buurman W, et al. (2014) Intestinal permeability-A new target for disease prevention and therapy. BMC Gastroenterol 14: 189–214. doi: 10.1186/s12876-014-0189-7
![]() |
[7] |
Van Itallie CM, Holmes J, Bridges A, et al. (2008) The density of small tight junction pores varies among cell types and is increased by expression of claudin-2. J Cell Sci 121: 298–305. doi: 10.1242/jcs.021485
![]() |
[8] |
Ulluwishewa D, Anderson RC, Mcnabb WC, et al. (2011) Regulation of tight junction permeability by intestinal bacteria and dietary components. J Nutr 141: 769–776. doi: 10.3945/jn.110.135657
![]() |
[9] | De Magistris L, Picardi A, Sapone A, et al. (2014) Intestinal barrier in autism, In: Patel VB (ed.), Comprehensive guide to autism, New York: Springer, 123. |
[10] |
Catassi C, Fasano A (2008) Celiac disease. Curr Opin Gastroenterol 24: 687–691. doi: 10.1097/MOG.0b013e32830edc1e
![]() |
[11] |
Bjarnason I, Macpherson A, Hollander D (1995) Intestinal permeability: An overview. Gastroenterology 108: 1566–1581. doi: 10.1016/0016-5085(95)90708-4
![]() |
[12] |
Fasano A (2011) Zonulin and its regulation of intestinal barrier function: The biological door to inflammation, autoimmunity and cancer. Physiol Rev 91: 151–175. doi: 10.1152/physrev.00003.2008
![]() |
[13] |
Lerner A, Matthias T (2015) Changes in intestinal tight junction permeability associated with industrial food additives explain the rising incidence of autoimmune disease. Autoimmun Rev 14: 479–489. doi: 10.1016/j.autrev.2015.01.009
![]() |
[14] |
Siniscalco D, Cirillo A, Bradstreet JJ, et al. (2013) Epigenetic findings in autism: New perspectives for therapy. Int J Environ Res Public Health 10: 4261–4273. doi: 10.3390/ijerph10094261
![]() |
[15] |
Siniscalco D, Antonucci N (2013) Possible use of Trichuris suis ova in autism spectrum disorders therapy. Med Hypotheses 81: 1–4. doi: 10.1016/j.mehy.2013.03.024
![]() |
[16] |
Wakefield AJ (2002) The gut-brain axis in childhood developmental disorders. J Pediatr Gastroenterol Nutr 34: S14–S17. doi: 10.1097/00005176-200205001-00004
![]() |
[17] | Siniscalco D (2014) Gut bacteria-brain axis in autism. Autism 4: e124. |
[18] | Siniscalco D, Antonucci N (2013) Involvement of dietary bioactive proteins and peptides in autism spectrum disorders. Curr Protein Pept Sci 14: 674–679. |
[19] |
Trivedi MS, Shah JS, Al-Mughairy S, et al. (2014) Food-derived opioid peptides inhibit cysteine uptake with redox and epigenetic consequences. J Nutr Biochem 25: 1011–1018. doi: 10.1016/j.jnutbio.2014.05.004
![]() |
[20] |
Frustaci A, Neri M, Cesario A, et al. (2012) Oxidative stress-related biomarkers in autism: Systematic review and meta-analyses. Free Radic Biol Med 52: 2128–2141. doi: 10.1016/j.freeradbiomed.2012.03.011
![]() |
[21] |
Melnyk S, Fuchs GJ, Schulz E, et al. (2012) Metabolic imbalance associated with methylation dys-regulation and oxidative damage in children with autism. J Autism Dev Disord 42: 367–377. doi: 10.1007/s10803-011-1260-7
![]() |
[22] |
Shattock P, Whiteley P (2002) Biochemical aspects in autism spectrum disorders: Updating the opioid-excess theory and presenting new opportunities for biomedical intervention. Expert Opin Ther Targets 6: 175–183. doi: 10.1517/14728222.6.2.175
![]() |
[23] |
Siniscalco D, Sapone A, Giordano C, et al. (2013) Cannabinoid receptor type 2, but not type 1, is up-regulated in peripheral blood mononuclear cells of children affected by autistic disorders. J Autism Dev Disord 43: 2686–2695. doi: 10.1007/s10803-013-1824-9
![]() |
[24] |
Siniscalco D, Bradstreet JJ, Cirillo A, et al. (2014) The in vitro GcMAF effects on endocannabinoid system transcriptionomics, receptor formation, and cell activity of autism-derived macrophages. J Neuroinflammation 11: 78. doi: 10.1186/1742-2094-11-78
![]() |
[25] |
Fiorentino M, Sapone A, Senger S, et al. (2016) Blood-brain barrier and intestinal epithelial barrier alterations in autism spectrum disorders. Mol Autism 7: 49. doi: 10.1186/s13229-016-0110-z
![]() |
[26] |
Rose DR, Yang H, Serena G, et al. (2018) Differential immune responses and microbiota profiles in children with autism spectrum disorders and co-morbid gastrointestinal symptoms. Brain Behav Immun 70: 354–368. doi: 10.1016/j.bbi.2018.03.025
![]() |
[27] |
Lionetti E, Leonardi S, Franzonello C, et al. (2015) Gluten psychosis: Confirmation of a new clinical entity. Nutrients 7: 5532–5539. doi: 10.3390/nu7075235
![]() |
[28] |
O'Mahony SM, Clarke G, Borre YE, et al. (2015) Serotonin, tryptophan metabolism and the brain-gut-microbiome axis. Behav Brain Res 277: 32–48. doi: 10.1016/j.bbr.2014.07.027
![]() |
[29] |
Theije CGMD, Koelink PJ, Korte-Bouws GA, et al. (2014) Intestinal inflammation in a murine model of autism spectrum disorders. Brain Behav Immun 37: 240–247. doi: 10.1016/j.bbi.2013.12.004
![]() |
[30] |
Baganz NL, Blakely RD (2013) A dialogue between the immune system and brain, spoken in the language of serotonin. ACS Chem Neurosci 4: 48–63. doi: 10.1021/cn300186b
![]() |
[31] |
Van Elst K, Bruining H, Birtoli B, et al. (2014) Food for thought: Dietary changes in essential fatty acid ratios and the increase in autism spectrum disorders. Neurosci Biobehav Rev 45: 369–738. doi: 10.1016/j.neubiorev.2014.07.004
![]() |
[32] |
Halliwell B (2006) Oxidative stress and neurodegeneration: Where are we now? J Neurochem 97: 1634–1658. doi: 10.1111/j.1471-4159.2006.03907.x
![]() |
[33] |
Cartocci V, Catallo M, Tempestilli M, et al. (2018) Altered brain cholesterol/isoprenoid metabolism in a rat model of autism spectrum disorders. Neuroscience 372: 27–37. doi: 10.1016/j.neuroscience.2017.12.053
![]() |
[34] |
Brigida AL, Schultz S, Cascone M, et al. (2017) Endocannabinod signal dysregulation in autism spectrum disorders: A correlation link between inflammatory state and neuro-immune alterations. Int J Mol Sci 18: 1425. doi: 10.3390/ijms18071425
![]() |
[35] |
Acharya N, Penukonda S, Shcheglova T, et al. (2017) Endocannabinoid system acts as a regulator of immune homeostasis in the gut. Proc Natl Acad Sci USA 114: 5005–5010. doi: 10.1073/pnas.1612177114
![]() |
[36] |
Gyires K, Zádori ZS (2016) Role of cannabinoids in gastrointestinal mucosal defense and inflammation. Curr Neuropharmacol 14: 935–951. doi: 10.2174/1570159X14666160303110150
![]() |
[37] |
Maríbauset S, Llopisgonzález A, Zazpe I, et al. (2016) Nutritional Impact of a Gluten-Free Casein-Free Diet in Children with Autism Spectrum Disorder. J Autism Dev Disord 46: 673–684. doi: 10.1007/s10803-015-2582-7
![]() |
[38] |
Iovene MR, Bombace F, Maresca R, et al. (2017) Intestinal Dysbiosis and Yeast Isolation in Stool of Subjects with Autism Spectrum Disorders. Mycopathologia 182: 349–363. doi: 10.1007/s11046-016-0068-6
![]() |
[39] | Siniscalco D, Mijatovic T, Bosmans E, et al. (2016) Decreased Numbers of CD57 + CD3- Cells Identify Potential Innate Immune Differences in Patients with Autism Spectrum Disorder. Vivo 30: 83–89. |
[40] |
de Theije CG, Wopereis H, Ramadan M, et al. (2014) Altered gut microbiota and activity in a murine model of autism spectrum disorders. Brain Behav Immun 37: 197–206. doi: 10.1016/j.bbi.2013.12.005
![]() |
[41] |
Needham BD, Tang W, Wu WL (2018) Searching for the gut microbial contributing factors to social behavior in rodent models of autism spectrum disorder. Dev Neurobiol 78: 474–499. doi: 10.1002/dneu.22581
![]() |
[42] |
Fung TC, Olson CA, Hsiao EY (2017) Interactions between the microbiota, immune and nervous systems in health and disease. Nat Neurosci 20: 145–155. doi: 10.1038/nn.4476
![]() |
[43] |
Santocchi E, Guiducci L, Fulceri F, et al. (2016) Gut to brain interaction in Autism Spectrum Disorders: A randomized controlled trial on the role of probiotics on clinical, biochemical and neurophysiological parameters. BMC Psychiatry 16: 183. doi: 10.1186/s12888-016-0887-5
![]() |
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