## Topological Methods in Nonlinear Analysis

### Attractors and global averaging of non-autonomous reaction-diffusion equations in $\mathbb R^N$

#### Abstract

We consider a family of non-autonomous reaction-diffusion equations $$u_t=\sum_{i,j=1}^N a_{ij}(\omega t)\partial_i\partial_j u+f(\omega t,u)+ g(\omega t,x), \quad x\in\mathbb R^N \tag{\text{\rm E}_\omega}$$ with almost periodic, rapidly oscillating principal part and nonlinear interactions. As $\omega\to \infty$, we prove that the solutions of $(\text{\rm E}_\omega)$ converge to the solutions of the averaged equation $$u_t=\sum_{i,j=1}^N \overline a_{ij}\partial_i\partial_j u+\overline f(u)+ \overline g(x), \quad x\in\mathbb R^N. \tag{\text{\rm E}_\infty}$$ If $f$ is dissipative, we prove existence and upper-semicontinuity of attractors for the family (E$_\omega$) as $\omega\to\infty$.

#### Article information

Source
Topol. Methods Nonlinear Anal., Volume 20, Number 2 (2002), 229-259.

Dates
First available in Project Euclid: 1 August 2016

https://projecteuclid.org/euclid.tmna/1470081174

Mathematical Reviews number (MathSciNet)
MR1962220

Zentralblatt MATH identifier
1039.35021

#### Citation

Antoci, Francesca; Prizzi, Martino. Attractors and global averaging of non-autonomous reaction-diffusion equations in $\mathbb R^N$. Topol. Methods Nonlinear Anal. 20 (2002), no. 2, 229--259. https://projecteuclid.org/euclid.tmna/1470081174

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