Asymptotic behavior for nonlinear degenerate parabolic equations with irregular data

ABSTRACT

This paper focuses on the following degenerate parabolic equation $\begin{array}{rl}& u_t-\mathrm{d}\mathrm{i}\mathrm{v}\left(\sigma \right(x\left)|\mathrm{\nabla }u{|}^{p-2}\mathrm{\nabla }u\right)+f(x,u)=g\mathrm{i}\mathrm{n}\mathrm{\Omega }\times {\mathbb{R}}^{+},\\ & u=0\mathrm{o}\mathrm{n}\mathrm{\partial }\mathrm{\Omega }\times {\mathbb{R}}^{+},\\ & u(x,0)=u_0\left(x\right)\mathrm{i}\mathrm{n}\mathrm{\Omega },\end{array}$ where Ω is a smooth bounded domain in ${\mathbb{R}}^N,(N\ge 2),1<p<N,u_0,g\in L^1\left(\mathrm{\Omega }\right)$, $\sigma \left(x\right)$ is positive almost everywhere and satisfies proper degenerate conditions. The existence and uniqueness result is proved in the framework of entropy solutions. For the long-time behavior, we prove the existence of a global attractor in $L^q\left(\mathrm{\Omega }\right)$ by using some delicate estimates on the solution, which are derived by taking advantage of both the leading operator and the zero-order nonlinear term. The aforementioned results improve some previous results in the literature in several aspects.

Keywords:

• Degenerate variable diffusion
• asymptotic behaviors
• irregular data

2010 Mathematics Subject Classifications:

• Primary 37L30
• 35B41
• Secondary35Q35

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was partially supported by the National Natural Science Foundation of China (NSFC) [grant number 11701002, 11971031].