Solvability of a doubly singular boundary value problem arising in front propagation for reaction-diffusion equations

TL;DR

This paper investigates the conditions under which a specific doubly singular boundary value problem, related to traveling wave solutions in reaction-diffusion equations with the p-Laplacian, is solvable.

Contribution

It provides new results on the existence and properties of solutions to a doubly singular boundary value problem in reaction-diffusion equations involving the p-Laplacian.

Findings

Established solvability conditions for the boundary value problem.

Analyzed the influence of parameters c and α on solution existence.

Connected the problem to traveling wave solutions in reaction-diffusion systems.

Abstract

The paper deals with the solvability of the following doubly singular boundary value problem \[\begin{cases} \dot z = c g(u)-f(u) -\dfrac{h(u)}{z^\alpha}\\ z(0^+)=0, z(1^-)=0, \ z(u)>0 \text{ in } (0,1)\end{cases}\] naturally arising in the study of the existence and properties of travelling waves for reaction-diffusion-convection equations governed by the $p-$Laplacian operator. Here $c,\alpha$ are real parameters, with $\alpha>0$, and $f,g,h$ are continuous functions in $[0,1]$, with \[ h(0)=h(1), \quad h(u)>0 \text{ in } (0,1).\]