# Nonlinear estimates for traveling wave solutions of reaction diffusion equations

## TL;DR

This paper develops nonlinear bounds for traveling wave solutions of reaction-diffusion equations and applies these bounds to the Lotka-Volterra system, extending previous linear maximum principle methods.

## Contribution

It introduces nonlinear a priori bounds for a broad class of reaction-diffusion equations, including the Lotka-Volterra system, using an extension of the linear N-barrier maximum principle.

## Key findings

- Established nonlinear bounds for solutions of reaction-diffusion equations.
- Applied bounds to the Lotka-Volterra system of two species.
- Extended linear maximum principle techniques to nonlinear estimates.

## Abstract

In this paper we will establish nonlinear a priori lower and upper bounds for the solutions to a large class of equations which arise from the study of traveling wave solutions of reaction-diffusion equations, and we will apply our nonlinear bounds to the Lotka-Volterra system of two competing species as examples. The idea used in a series of papers \cite{NBMP-Discrete,JDE-16,CPAA-16,DCDS-B-18,NBMP-n-species,DCDS-A-17} for the establishment of the linear N-barrier maximum principle will also be used in the proof.