Dynamical behavior of a degenerate parabolic equation with memory on the whole space

Abstract

This paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on $\mathbb{R}^{n}$ R n . Since the corresponding equation includes the degenerate term $\operatorname{div}\{a(x)\nabla u\}$ div { a ( x ) ∇ u } , it requires us to give appropriate assumptions about the weight function $a(x)$ a ( x ) for studying our problem. Based on this, we first obtain the existence of a bounded absorbing set, then verify the asymptotic compactness of a solution semigroup via the asymptotic contractive semigroup method. Finally, the existence and uniqueness of global attractors are proved. In particular, the nonlinearity f satisfies the polynomial growth of arbitrary order $p-1$ p − 1 ($p\geq 2$ p ≥ 2 ) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions.