Selfsimilar solutions in a sector for a quasilinear parabolic equation

TL;DR

This paper investigates self-similar solutions in a sector for a quasilinear parabolic equation, establishing existence, uniqueness, and stability of both expanding and shrinking solutions that are self-similar at discrete times.

Contribution

It introduces a novel analysis of self-similar solutions in a sector for quasilinear parabolic equations, including proofs of existence, uniqueness, and stability.

Findings

Existence of expanding self-similar solutions

Uniqueness of shrinking self-similar solutions

Asymptotic stability of the solutions

Abstract

We study a two-point free boundary problem in a sector for a quasilinear parabolic equation. The boundary conditions are assumed to be spatially and temporally "self-similar" in a special way. We prove the existence, uniqueness and asymptotic stability of an expanding solution which is self-similar at discrete times. We also study the existence and uniqueness of a shrinking solution which is self-similar at discrete times.