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

Electronic Journal of Differential Equations

卷:2016

Jin Zhang¹  Chang Zhang²  Chengkui Zhong² 

[1] Hohai Univ., Nanjing, China ;

[2] Nanjing Univ., Nanjing, China ;

关键词: Global attractor;    inhomogeneous reaction-diffusion equation;    unbounded domain;    supercritical nonlinearity;   

DOI  :  

来源: DOAJ

【 摘 要 】

We consider the existence of global attractor for theinhomogeneous 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 $L^2(\mathbb{R}^n)\cap L^p(\mathbb{R}^n)$.

【 授权许可】

Unknown