Advances in Differential Equations

The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity

M. A. Efendiev, H. Gajewski, and S. Zelik

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Abstract

This article is devoted to the study of the long-time behavior of solutions of the following 4th order parabolic system in a bounded smooth domain $\Omega\subset\subset{\mathbb R}^n$: $$ b\partial_t u=-{\Delta_x} (a{\Delta_x} u-\alpha\partial_t u-f(u)+\tilde g )\,, \tag*{(1)} $$ where $u=(u^1,\cdots,u^k)$ is an unknown vector-valued function, $a$ and $b$ are given constant matrices such that $a+a^*>0$, $b=b^*>0$, $\alpha>0$ is a positive number, and $f$ and $g$ are given functions. Note that the nonlinearity $f$ is not assumed to be subordinated to the Laplacian. The existence of a finite dimensional global attractor for system (1) is proved under some natural assumptions on the nonlinear term $f$.

Article information

Source
Adv. Differential Equations Volume 7, Number 9 (2002), 1073-1100.

Dates
First available in Project Euclid: 29 April 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1367241460

Mathematical Reviews number (MathSciNet)
MR1920373

Zentralblatt MATH identifier
1035.35019

Subjects
Primary: 35B41: Attractors
Secondary: 35K25: Higher-order parabolic equations 35K50 35K55: Nonlinear parabolic equations 37L30: Attractors and their dimensions, Lyapunov exponents

Citation

Efendiev, M. A.; Gajewski, H.; Zelik, S. The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity. Adv. Differential Equations 7 (2002), no. 9, 1073--1100. https://projecteuclid.org/euclid.ade/1367241460.


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