Asymptotic decay towards steady states of solutions to very fast and singular diffusion equations

TL;DR

This paper investigates the long-term behavior of solutions to very fast and singular diffusion equations with nonhomogeneous sources, identifying critical parameters for convergence to steady states in 2D and 3D.

Contribution

It determines critical values of the porous medium exponent ensuring convergence to steady states in specific dimensions, advancing understanding of these diffusion processes.

Findings

Critical exponent values for asymptotic convergence in 2D and 3D

Conditions for solutions to approach unique steady states

Analysis of decay rates towards steady states

Abstract

We analyze long-time behavior of solutions to a class of problems related to very fast and singular diffusion porous medium equations having nonhomogeneous in space and time source terms with zero mean. In dimensions two and three, we determine critical values of porous medium exponent for the asymptotic $H^1$-convergence of the solutions to a unique nonhomogeneous positive steady state generally to hold.