Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents

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• 作者：Peter E. Kloeden^a ; ^{kloeden@math.uni-frankfurt.de" class="auth_mail" title="E-mail the corresponding author} ; Jacson Simsen^b ; ^c ; ^{jacson@unifei.edu.br" class="auth_mail" title="E-mail the corresponding author} ; Mariza Stefanello Simsen^b ; ^c ; ^{mariza@unifei.edu.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Multivalued Cauchy problem ; Variable exponents ; Pullback attractors ; Time-dependent operator ; Asymptotically autonomous inclusion
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：513-531
• 全文大小：414 K

文摘

We study the asymptotic behavior of a non-autonomous multivalued Cauchy problem of the form on a bounded smooth domain Ω in Rⁿden">de">${\mathbb{R}}^n$, n≥1den">de">$n\ge 1$ with a homogeneous Neumann boundary condition, where the exponent den">de">$p(\cdot )\in C(\bar{\mathrm{\Omega }})$ satisfies p⁻den">de">$p^{-}$ := min⁡p(x)>2den">de">$\min p(x)>2$. We prove the existence of a pullback attractor and study the asymptotic upper semicontinuity of the elements of the pullback attractor A={A(t):t∈R}den">de">$\mathfrak{A}=\{\mathcal{A}(t):t\in \mathbb{R}\}$ as t→∞den">de">$t\to \infty$ for the non-autonomous evolution inclusion in a Hilbert space H under the assumptions, amongst others, that F   is a measurable multifunction and D∈L^∞([τ,T]×Ω)den">de">$D\in L^{\infty }([\tau ,T]\times \mathrm{\Omega })$ is bounded above and below and is monotonically nonincreasing in time. The global existence of solutions is obtained through results of Papageorgiou and Papalini.