Well-Posedness of Initial-Boundary Value Problems for a Reaction-Diffusion Equation

TL;DR

This paper proves local well-posedness for a reaction-diffusion equation with power nonlinearity on half-line or finite interval, using a novel approach based on the unified transform method to obtain linear estimates.

Contribution

It extends the unified transform method approach from dispersive to diffusive equations, establishing well-posedness for reaction-diffusion problems with boundary conditions.

Findings

Well-posedness in Sobolev spaces for reaction-diffusion equations.

Application of the unified transform method to diffusive equations.

Linear estimates analogous to dispersive equations derived.

Abstract

A reaction-diffusion equation with power nonlinearity formulated either on the half-line or on the finite interval with nonzero boundary conditions is shown to be locally well-posed in the sense of Hadamard for data in Sobolev spaces. The result is established via a contraction mapping argument, taking advantage of a novel approach that utilizes the formula produced by the unified transform method of Fokas for the forced linear heat equation to obtain linear estimates analogous to those previously derived for the nonlinear Schr\"odinger, Korteweg-de Vries and "good" Boussinesq equations. Thus, the present work extends the recently introduced "unified transform method approach to well-posedness" from dispersive equations to diffusive ones.