The One-Dimensional Nonlinear Heat Equation with Absorption: Regularity of Solutions and Interfaces

Abstract

We consider the equation $u_t = (u^m )_{xx} - \lambda u^n $ with $m > 1$, $\lambda > 0$, $n \geqq m$ as a model for heat diffusion with absorption. Hence we assume that $u \geqq 0$ for $x \in \mathbb{R}$, $t \geqq 0$. We study the regularity of the solution to the Cauchy problem for this degenerate parabolic equation. When the initial datum $u_0 (x)$ is positive only in a part of the space $\mathbb{R}$, we also study the regularity of the free boundaries that appear. The asymptotic behavior of solutions and free boundaries is also discussed.