Front propagation in nonlinear parabolic equations

TL;DR

This paper investigates the existence and stability of traveling wave solutions in nonlinear convection-diffusion equations with gradient-dependent diffusion, including degenerate cases, demonstrating unconditional stability in Lebesgue spaces.

Contribution

It introduces new results on the existence and unconditional stability of traveling waves for nonlinear equations with gradient-dependent, possibly degenerate diffusion.

Findings

Existence of traveling waves established.

Unconditional stability proven in Lebesgue spaces.

Applicable to degenerate diffusion scenarios.

Abstract

We study existence and stability of travelling waves for nonlinear convection diffusion equations in the 1-D Euclidean space. The diffusion coefficient depends on the gradient in analogy with the p-Laplacian and may be degenerate. Unconditional stability is established with respect to initial data perturbations in the Lebesgue space of integrable functions.