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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304024&_mathId=si2.gif&_user=111111111&_pii=S0022247X16304024&_rdoc=1&_issn=0022247X&md5=1409285b255bbc155b3cf510936386d6" title="Click to view the MathML source">Rⁿclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^n$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304024&_mathId=si3.gif&_user=111111111&_pii=S0022247X16304024&_rdoc=1&_issn=0022247X&md5=dd58b60303ca9359a6219eef7330769a" title="Click to view the MathML source">n≥1class="mathContainer hidden">class="mathCode">$n\ge 1$ with a homogeneous Neumann boundary condition, where the exponent class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304024&_mathId=si4.gif&_user=111111111&_pii=S0022247X16304024&_rdoc=1&_issn=0022247X&md5=f99a7a442b68bb782a7b433f1aa910c2">class="imgLazyJSB inlineImage" height="19" width="84" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16304024-si4.gif">class="mathContainer hidden">class="mathCode">$p(\cdot )\in C(\bar{\mathrm{\Omega }})$ satisfies class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304024&_mathId=si5.gif&_user=111111111&_pii=S0022247X16304024&_rdoc=1&_issn=0022247X&md5=25c81e04e34e77882857042a6c3ebbe1" title="Click to view the MathML source">p⁻class="mathContainer hidden">class="mathCode">$p^{-}$ := class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304024&_mathId=si6.gif&_user=111111111&_pii=S0022247X16304024&_rdoc=1&_issn=0022247X&md5=c2e10304e3b1c9786693b573ab9bfbe8" title="Click to view the MathML source">min⁡p(x)>2class="mathContainer hidden">class="mathCode">$\min p(x)>2$. We prove the existence of a pullback attractor and study the asymptotic upper semicontinuity of the elements of the pullback attractor class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304024&_mathId=si276.gif&_user=111111111&_pii=S0022247X16304024&_rdoc=1&_issn=0022247X&md5=bcdca4f5d89991204805333e3a7988e9" title="Click to view the MathML source">A={A(t):t∈R}class="mathContainer hidden">class="mathCode">$\mathfrak{A}=\{\mathcal{A}(t):t\in \mathbb{R}\}$ as class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304024&_mathId=si338.gif&_user=111111111&_pii=S0022247X16304024&_rdoc=1&_issn=0022247X&md5=8d39b0a75c6c1580f98e0a231b40265b" title="Click to view the MathML source">t→∞class="mathContainer hidden">class="mathCode">$t\to \infty$ for the non-autonomous evolution inclusion in a Hilbert space H under the assumptions, amongst others, that F   is a measurable multifunction and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16304024&_mathId=si9.gif&_user=111111111&_pii=S0022247X16304024&_rdoc=1&_issn=0022247X&md5=a8ce5998a47a07098c08992361c049b4" title="Click to view the MathML source">D∈L^∞([τ,T]×Ω)class="mathContainer hidden">class="mathCode">$D\in L^{\infty }([\tau ,T]\times \mathrm{\Omega })$ is bounded above and below and is monotonically nonincreasing in time. The global existence of solutions is obtained through results of Papageorgiou and Papalini.