Singular integral equations with applications to travelling waves for doubly nonlinear diffusion

TL;DR

This paper analyzes singular Volterra integral equations related to travelling-wave solutions in doubly nonlinear diffusion equations, extending previous results for the case p=2 and developing new analytical tools.

Contribution

It extends existing analysis of integral equations to the p≠2 case, enabling the study of travelling waves in more general nonlinear diffusion-reaction models.

Findings

Extended analysis to p≠2 case introduces new singularity handling techniques.

Established existence and properties of travelling-wave solutions for doubly nonlinear diffusion.

Developed new mathematical tools for analyzing singular integral equations.

Abstract

We consider a family of singular Volterra integral equations that appear in the study of monotone travelling-wave solutions for a family of diffusion-convection-reaction equations involving the $p$-Laplacian operator. Our results extend the ones due to B.\,Gilding for the case $p=2$. The fact that $p\neq2$ modifies the nature of the singularity in the integral equation, and introduces the need to develop some new tools for the analysis. The results for the integral equation are then used to study the existence and properties of travelling-wave solutions for doubly nonlinear diffusion-reaction equations in terms of the constitutive functions of the problem.