Front-like entire solutions for reaction-diffusion equations with convection

Abstract

This talk is concerned with families of front-like entire solutions for problems with convection, both for bistable and monostable reaction-diffusion-convection equations, and, via vanishing-viscosity arguments, for bistable and monostable  balance laws. The unified approach employed  is  based on a robust method using front-dependent  sub and supersolutions. Unlike convectionless problems, the equations studied here lack symmetry between increasing and decreasing travelling waves, which affects  the  choice of sub and supersolutions used.  Our entire solutions  include both those that behave like two fronts coming together and annihilating as time increases, and, for bistable equations,  those that behave like two fronts merging to propagate like a single front. We also characterise the long-time  behaviour of each family of entire solutions, which in the case of solutions constructed from a monostable front merging with a bistable front answers a question that was open even for reaction-diffusion equations without convection. This is joint work with Je-Chiang Tsai, National Chung Cheng University, Taiwan.