Electron. J. Diff. Equ., Vol. 2016 (2016), No. 63, pp. 1-9.

Global attractor for reaction-diffusion equations with supercritical nonlinearity in unbounded domains

Jin Zhang, Chang Zhang, Chengkui Zhong

Abstract:
We consider the existence of global attractor for the inhomogeneous reaction-diffusion equation
$$\displaylines{ u_t- \Delta u - V(x)u + |u|^{p-2}u =g, \quad \text{in } \mathbb{R}^n\times\mathbb{R}^{+},\cr u(0) = u_0\in L^2(\mathbb{R}^n)\cap L^p(\mathbb{R}^n), }$$
where $p>\frac{2n}{n-2}$ is supercritical and V(x) satisfies suitable assumptions. Since $-\Delta$ is not positive definite in $H^1(\mathbb{R}^n)$, the Gronwall inequality can not be derived and the corresponding semigroup does not possess bounded absorbing sets in $L^2(\mathbb{R}^n)$. Thus, by a special method, we prove that the equation has a global attractor in $L^p(\mathbb{R}^n)$, which attracts any bounded subset in $L^2(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$.

Submitted July 20, 2015. Published March 4, 2016.
Math Subject Classifications: 35B41, 35K57, 37L99.
Key Words: Global attractor; inhomogeneous reaction-diffusion equation; unbounded domain; supercritical nonlinearity.

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Jin Zhang
Department of Mathematics, College of Science
Hohai University
Nanjing 210098, China
email: zhangjin86@hhu.edu.cn
Chang Zhang
Department of Mathematics, Nanjing University
Nanjing 210093, China
email: chzhnju@126.com
Chengkui Zhong
Department of Mathematics, Nanjing University
Nanjing 210093, China
email: ckzhong@nju.edu.cn

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