Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption

TL;DR

This paper investigates a nonlinear diffusion equation with strong spatially inhomogeneous absorption, proving support shrinking, existence of self-similar solutions, and non-extinction for a broad class of initial conditions.

Contribution

It establishes support shrinking and localization, constructs a unique self-similar solution, and shows non-extinction for solutions with inhomogeneous absorption.

Findings

Supports shrink instantaneously for all positive times.

Existence of a unique, radially symmetric self-similar solution.

Solutions do not extinguish in finite time for broad initial conditions.

Abstract

We study the dynamics of the following porous medium equation with strong absorption $$\partial_t u=\Delta u^m-|x|^{\sigma}u^q,$$ posed for $(t, x) \in (0,\infty) \times \mathbb{R}^N$, with $m > 1$, $q \in (0, 1)$ and $\sigma > 2(1-q)/(m-1)$. Considering the Cauchy problem with non-negative initial condition $u_0 \in L^\infty(\mathbb{R}^N)$ instantaneous shrinking and localization of supports for the solution $u(t)$ at any $t > 0$ are established. With the help of this property, existence and uniqueness of a nonnegative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.