Asymptotic behavior for the fast diffusion equation with absorption and singularity

TL;DR

This paper investigates the existence, decay, and asymptotic behavior of solutions to a fast diffusion equation with absorption and singularity, using entropy methods to show convergence to a Barenblatt solution.

Contribution

It establishes the existence, decay estimates, and asymptotic convergence of weak solutions for the fast diffusion equation with absorption and singularity, addressing technical challenges from spatial singularities.

Findings

Proved existence and decay estimates for weak solutions.

Showed convergence to Barenblatt solutions under specified conditions.

Applied entropy dissipation and inequalities to analyze asymptotic behavior.

Abstract

This paper is concerned with the weak solution for the fast diffusion equation with absorption and singularity in the form of $u_t=\triangle u^m -u^p$. We first prove the existence and decay estimate of weak solution when the fast diffusion index satisfies $0<m<1$ and the absorption index is $p>1$. Then we show the asymptotic convergence of weak solution to the corresponding Barenblatt solution for $\frac{n-1}{n}<m<1$ and $p>m+\frac{2}{n}$ via the entropy dissipation method combining the generalized Shannon's inequality and Csisz$\mathrm{\acute{a}}$r-Kullback inequality. The singularity of spatial diffusion causes us the technical challenges for the asymptotic behavior of weak solution.