Global attractor for the m-semiflow generated by a quasilinear degenerate parabolic equation

Abstract

Using theory of global attractors for multi-valued semiflows, we prove the existence of a global attractor for the m-semiflow generated by a parabolic equation involving the nonlinear degenerate operator in a bounded domain.

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

“…Using the Gronwall lemma, there exists a constant T 0 ( u 0 H 1,a 0 ∩L p ), such that u(t) 2 2 ≤ ρ 0 for t ≥ T 0 . Taking t ≥ T 0 and integrating (11)…”

Pullback attractor for differential evolution inclusions with infinite delays

“…Thus F (t, K ) = F 0 (t, K ) + F 1 (K) is a relatively compact set, and then F is quasicompact. To verify (F) (2), it suffices to show that F(t, ·) is closed. Let {φ n } be a sequence in C γ converging to φ * and ξ n ∈ F(t, φ n ) be such that ξ n → ξ * .…”

“…Thanks to the theories of attractors for multivalued semiflows/processes given in [7,8,10,20,21], one can find a global attractor for semiflows/processes governed by solutions of DIs, which is a compact set attracting all solutions as the time goes to infinity in some contexts. Some deployments of these theories for differential inclusions and differential equations without uniqueness can be found in [2,3,[8][9][10]18,24,25].…”

LONG-TIME BEHAVIOR FOR SEMILINEAR DEGENERATE PARABOLIC EQUATIONS ON ℝN

“…[1,4,10,11,12]) and quasilinear case (cf. [2,3,5,6]). However, all of the above results are in the compact case, that is the case where the weights were assumed to satisfy certain conditions which ensure the compactness of some Sobolev embeddings, and this plays an essential role in the study of these works.…”