Self-Similar Intermediate Asymptotics for a Degenerate Parabolic Filtration-Absorption Equation

TL;DR

This paper develops self-similar solutions for a degenerate parabolic filtration-absorption equation modeling groundwater flow in fissurized porous rock, and investigates their role as intermediate asymptotics through numerical experiments.

Contribution

It constructs a family of self-similar solutions and demonstrates their significance as intermediate asymptotics for a broader class of solutions.

Findings

Self-similar solutions act as intermediate asymptotics.

Numerical experiments confirm the relevance of these solutions.

Discussion of nonuniqueness issues in the Cauchy problem.

Abstract

The equation $$ \partial_tu=u\partial^2_{xx}u-(c-1)(\partial_xu)^2 $$ is known in literature as a qualitative mathematical model of some biological phenomena. Here this equation is derived as a model of the groundwater flow in a water absorbing fissurized porous rock, therefore we refer to this equation as a filtration-absorption equation. A family of self-similar solutions to this equation is constructed. Numerical investigation of the evolution of non-self-similar solutions to the Cauchy problems having compactly supported initial conditions is performed. Numerical experiments indicate that the self-similar solutions obtained represent intermediate asymptotics of a wider class of solutions when the influence of details of the initial conditions disappears but the solution is still far from the ultimate state: identical zero. An open problem caused by the nonuniqueness of the solution of the Cauchy problem is discussed.