Attractors for a class of semi-linear degenerate parabolic equations

Abstract

We consider degenerate parabolic equations of the form ∂tu = Δλu + f(u) u|∂Ω = 0, u|t=0 = u0 in a bounded domain Ω ⊂N, where Δλ is a subelliptic operator of the type (Formula presented.) We prove global existence of solutions and characterize their longtime behavior. In particular, we show the existence and finite fractal dimension of the global attractor of the generated semigroup and the convergence of solutions to an equilibrium solution when time tends to infinity

Attractors met X-elliptic operators

“…This generalizes our previous result in [23], where we studied Problem (1.1) for the particular class of ∆ λ -Laplacians (see Subsection 2.1.2).…”

“…However, for the ∆ λ -Laplacian the Poincaré inequality (P ) can also be directly verified as in A, and the Sobolev embeddings (S) were proved in [22] and [14]. If we apply Theorem 1 to the ∆ λ -Laplacian we recover our previous result in [23].…”

“…The aim of this article is to show that our previous result in [23] extends to a large class of parabolic equations involving degenerate elliptic operators and that similar results can be obtained for damped hyperbolic equations. Moreover, we formulate for the different classes of X-elliptic operators the admissible growth restrictions on the nonlinearity, which are determined by Sobolev type embedding theorems.…”

“…In Section 3 we collect some notions and results from the theory of semigroups and infinite dimensional dynamical systems that are used in the subsequent sections. Theorem 1 can be shown by following the arguments in [23]. For convenience of the reader we give a sketch of the proof in Section 4 and indicate the main ideas.…”

“…The local well-posedness of Problem (1.1) and Theorem 1 can be shown by slightly extending the arguments in [23]; for convenience of the reader we present a sketch of the proof and summarize the main ideas.…”