Traveling waves in nonclassical diffusion equations

TL;DR

This paper investigates the existence of monotone traveling wave solutions in nonclassical diffusion equations that combine standard diffusion with higher-order dispersive effects, employing fixed point methods and explicit super/subsolutions.

Contribution

It introduces a novel approach using less smooth super and subsolutions within a monotone iterative scheme to establish existence results for these complex equations.

Findings

Existence of monotone traveling wave solutions is proven.

Explicit super and subsolutions are constructed.

The method applies to equations with nonlinear reaction terms.

Abstract

We study the existence of monotone traveling wave solutions in a class of nonclassical diffusion equations that include both standard diffusion and a higher-order mixed space-time dispersive term. The reaction term is nonlinear and subject to general structural conditions. By employing the method of upper and lower solutions, using less smooth super and subsolutions, we construct a monotone iterative scheme within a convex set and prove its convergence using Schauder's fixed point theorem. Explicit constructions of super and subsolutions are provided.