# A heteroclinic orbit connecting traveling waves pertaining to different nonlinearities

## TL;DR

This paper proves the existence of a heteroclinic orbit connecting traveling waves associated with different nonlinearities in a semilinear parabolic equation, demonstrating convergence between solutions in an infinite cylinder.

## Contribution

It introduces a novel analysis of heteroclinic orbits connecting traveling waves for different spatially varying nonlinearities in parabolic equations.

## Key findings

- Existence of heteroclinic orbit connecting traveling waves
- Convergence of solutions to prescribed traveling waves
- Analysis in an infinite cylindrical domain

## Abstract

In this paper we consider a semilinear parabolic equation in an infinite cylinder. The spatially varying nonlinearity is such that it connects two (spatially independent) bistable nonlinearities in a compact set in space. We prove that, given such a setting, a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the cylinder, will converge to a traveling wave solution prescribed by the nonlinearity on the other side.