On the evolution -Laplacian with nonlocal memory

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• 作者：Stanislav Antontsev^a ; ^b ; ^{antontsevsn@mail.ru" class="auth_mail" title="E-mail the corresponding author} ; Sergey Shmarev^c ; ^{shmarev@uniovi.es" class="auth_mail" title="E-mail the corresponding author} ; Jacson Simsen^d ; ^{jacson@unifei.edu.br" class="auth_mail" title="E-mail the corresponding author} ; Mariza S. Simsen^d ; ^{mariza@unifei.edu.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35K92 ; 45K05 ; 35B99
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：March 2016
• 年：2016
• 卷：134
• 期：Complete
• 页码：31-54
• 全文大小：692 K

文摘

We study the homogeneous Dirichlet problem for the evolution p$p$-Laplacian with the nonlocal memory term where Ω⊂Rⁿ$\Omega \subset {\mathbb{R}}^n$ is a bounded domain, Θ$\Theta$, g$g$ and f$f$ are given functions. It is proved that for $max\{1,\frac{2n}{n+2}\}<p<\infty$, g,g^′∈L²(0,T)$g,g^{\prime }\in L^2(0,T)$ and $u_0\in W_0^{1,p}\left(\Omega \right)$, f∈L²(Q)$f\in L^2\left(Q\right)$ the problem admits a weak solution, which is global or local in time in dependence on the growth rate of Θ(x,t,s)$\Theta (x,t,s)$ as |s|→∞$\left|s\right|\to \infty$. Conditions of uniqueness are established. It is proved that for p>2$p>2$ and sΘ(x,t,s)≤0$s\Theta (x,t,s)\le 0$ the disturbances from the data propagate with finite speed and the “waiting time” effect is possible. We present simple explicit solutions that show the failure of the maximum and comparison principles for the solutions of equation (0.1).