Dynamical behavior of a degenerate parabolic equation with memory on the whole space. (English) Zbl 1537.35093
Summary: This paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on \(\mathbb{R}^n\). Since the corresponding equation includes the degenerate term \(\operatorname{div} \{a(x)\nabla u\}\), it requires us to give appropriate assumptions about the weight function \(a(x)\) 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 \(p-1\) \((p \geq 2)\) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions.
MSC:
| 35B41 | Attractors |
| 35K15 | Initial value problems for second-order parabolic equations |
| 35K58 | Semilinear parabolic equations |
| 35K65 | Degenerate parabolic equations |
Keywords:
degenerate parabolic equation; global attractor; contractive semigroup; memory; arbitrary polynomial growthReferences:
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