Long time existence of Smooth solution for the porous medium equation in a bounded domain

TL;DR

This paper proves the long-term existence of smooth solutions for the porous medium equation in bounded domains, utilizing estimates and regularity results to extend solutions over infinite time.

Contribution

It establishes the long-time existence of smooth solutions for the porous medium equation in bounded domains, based on advanced regularity and estimate techniques.

Findings

Proved long-time existence of smooth solutions.

Developed estimates for degenerate and mixed type equations.

Extended regularity results to porous medium equations.

Abstract

In this paper, we are going to show the long time existence of the smooth solution for the porous medium equations in a smooth bounded domain: {equation} {cases} u_t=\La u^m\quad\text{in $\Omega\times [0,\infty)$} u(x,0)=u_0>0\quad\text{in $\Omega$} u(x,t)=0\quad\text{for $x\in\partial\Omega$} {cases} {equation} where $m>1$ is the permeability. The proof is based on the short time existence of $C^{2,\bar{\gamma}}_{s}$-smooth solution, the global $C^{1}_{s}$-estimate, the H\"older estimate of divergence type degenerate equation with measurable coefficients and $C^{1,\bar{\gamma}}_s$-estimate of mixed type equation with Lipschitz coefficients.