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arXiv · 2405.07150

Asymptotic behavior for the fast diffusion equation with absorption and singularity

Changping Xie·Shaomei Fang·Ming Mei·Yuming Qin
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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 $0 1$. Then we show the asymptotic convergence of weak solution to the corresponding Barenblatt solution for $\frac{n-1}{n} m+\frac{2}{n}$ 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.

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Changping Xie, Shaomei Fang, Ming Mei, Yuming Qin. 2024-05-12. Asymptotic behavior for the fast diffusion equation with absorption and singularity. https://arxiv.org/abs/2405.07150

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