Long-time behavior for a class of weighted equations with degeneracy

Shan Ma; Chunyou Sun

doi:10.3934/dcds.2020098

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2020, Volume 40, Issue 3: 1889-1902. Doi: 10.3934/dcds.2020098

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Long-time behavior for a class of weighted equations with degeneracy

Shan Ma and
Chunyou Sun

School of Mathematics and Statistics, Lanzhou University, Lanzhou 730000, China

Received: June 30, 2019

Published: December 2019

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Abstract

In this paper we study the existence and some properties of the global attractors for a class of weighted equations when the weighted Sobolev space $ H_0^{1,a}(\Omega) $ (see Definition 1.1) cannot be bounded embedded into $ L^2(\Omega) $. We claim that the dimension of the global attractor is infinite by estimating its lower bound of $ Z_2 $-index. Moreover, we prove that there is an infinite sequence of stationary points in the global attractor which goes to 0 and the corresponding critical value sequence of the Lyapunov functional also goes to 0.

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Mathematics Subject Classification: Primary: 35K65, 35P30, 35B41.

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