Traveling waves for monostable reaction-diffusion-convection equations with discontinuous density-dependent coefficients

TL;DR

This paper develops a new framework for analyzing wave propagation in reaction-diffusion-convection equations with discontinuous and singular coefficients, introducing generalized traveling wave solutions and exploring their existence and properties.

Contribution

It introduces a novel concept of non-smooth traveling wave profiles for equations with discontinuous diffusion and convection, expanding the understanding of wave solutions in complex media.

Findings

Established conditions for existence and non-existence of generalized traveling waves.

Analyzed how convection influences the minimal wave speed.

Provided asymptotic behavior of wave profiles under power-type assumptions.

Abstract

This paper concerns wave propagation in a class of scalar reaction-diffusion-convection equations with $p$-Laplacian-type diffusion and monostable reaction. We introduce a new concept of a non-smooth traveling wave profile, which allows us to treat discontinuous diffusion with possible degenerations and singularities at 0 and 1, as well as only piecewise continuous convective velocity. Our approach is based on comparison arguments for an equivalent non-Lipschitz first-order ODE. We formulate sufficient conditions for the existence and non-existence of these generalized solutions and discuss how the convective velocity affects the minimal wave speed compared to the problem without convection. We also provide brief asymptotic analysis of the profiles, for which we need to assume power-type behavior of the diffusion and reaction terms.