Asymptotically autonomous multivalued Cauchy problems with spatially variable exponents

作者：Kloeden, Peter E.; Simsen, Jacson^*; Simsen, Mariza Stefanello

来源：Journal of Mathematical Analysis and Applications, 2017, 445(1): 513-531.

DOI：10.1016/j.jmaa.2016.08.004

• Multivalued Cauchy problem
• Variable exponents
• Pullback attractors
• Time-dependent operator
• Asymptotically autonomous inclusion

We study the asymptotic behavior of a non-autonomous multivalued Cauchy problem of the form
partial derivative u/partial derivative t(t) - div(D(t)vertical bar del u(t)vertical bar(p(x)-2)del u(t)) + vertical bar u(t)vertical bar(p(x)-2)u(t) + F(t, u(t)) (sic) 0
on a bounded smooth domain Omega in R-n >= 1 with a homogeneous Neumann boundary condition, where the exponent p(.) is an element of C((Omega) over bar) satisfies p- := 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 U = {A(t) : t is an element of R} as t -> infinity for the non -autonomous evolution inclusion in a Hilbert space H under the assumptions, amongst others, that F is a measurable multifunction and D is an element of L-infinity([tau,T] x 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.

• 出版日期2017-1-1
• 单位华中科技大学