Self-similarity and long-time behavior of solutions of the diffusion equation with nonlinear absorption and a boundary source

TL;DR

This paper investigates the long-time behavior of nonlinear reaction-diffusion equations modeling morphogen gradients, establishing the existence and uniqueness of ultra-singular self-similar solutions as limits of boundary-driven problems.

Contribution

It introduces a new class of ultra-singular self-similar solutions for nonlinear diffusion equations with boundary sources, proving their existence, uniqueness, and role as long-time limits.

Findings

Existence of ultra-singular self-similar solutions.

Uniqueness of these solutions in weighted energy spaces.

Self-similar solutions describe long-time behavior of the system.

Abstract

This paper deals with the long-time behavior of solutions of nonlinear reaction-diffusion equations describing formation of morphogen gradients, the concentration fields of molecules acting as spatial regulators of cell differentiation in developing tissues. For the considered class of models, we establish existence of a new type of ultra-singular self-similar solutions. These solutions arise as limits of the solutions of the initial value problem with zero initial data and infinitely strong source at the boundary. We prove existence and uniqueness of such solutions in the suitable weighted energy spaces. Moreover, we prove that the obtained self-similar solutions are the long-time limits of the solutions of the initial value problem with zero initial data and a time-independent boundary source.