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

Abstract

Abstract This paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on R n $\mathbb{R}^{n}$ . Since the corresponding equation includes the degenerate term div { a ( x ) ∇ u } $\operatorname{div}\{a(x)\nabla u\}$ , 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 ≥ 2 $p\geq 2$ ) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions.

Keywords

• Degenerate parabolic equation
• Global attractor
• Contractive semigroup
• Memory
• Arbitrary polynomial growth