# Exact travelling solution for a reaction-diffusion system with a piecewise constant production supported by a codimension-1 subspace

## TL;DR

This paper derives an exact monotone travelling wave solution for a reaction-diffusion system with production confined to a surface, relevant for biological modeling of processes like gene spread and morphogenesis.

## Contribution

It introduces a novel exact solution for a reaction-diffusion system with piecewise constant production on a codimension-1 subset, extending classical models.

## Key findings

- Solution propagates with a unique constant velocity
- Connects trivial and nontrivial stationary states
- Properties resemble classical bistable reaction-diffusion solutions

## Abstract

A generalisation of reaction diffusion systems and their travelling solutions to cases when the productive part of the reaction happens only on a surface in space or on a line on plane but the degradation and the diffusion happen in bulk are important for modelling various biological processes. These include problems of invasive species propagation along boundaries of ecozones, problems of gene spread in such situations, morphogenesis in cavities, intracellular reaction etc. Piecewise linear approximations of reaction terms in reaction-diffusion systems often result in exact solutions of propagation front problems. This article presents an exact travelling solution for a reaction-diffusion system with a piecewise constant production restricted to a codimension-1 subset. The solution is monotone, propagates with the unique constant velocity, and connects the trivial solution to a nontrivial nonhomogeneous stationary solution of the problem. The properties of the solution closely parallel the properties of monotone travelling solutions in classical bistable reaction-diffusion systems.