# Global existence and decay in nonlinearly coupled reaction-diffusion-advection equations with different velocities

## TL;DR

This paper proves global existence and decay for nonlinearly coupled reaction-diffusion-advection equations with different velocities, showing that certain mix-terms are harmless and providing conditions for long-term stability.

## Contribution

It introduces techniques to analyze the effect of transport on long-term dynamics, demonstrating global existence and decay in systems with different velocities and specific nonlinear terms.

## Key findings

- Quadratic or cubic mix-terms are harmless if components have different velocities.
- Global existence and Gaussian decay are established for localized initial data.
- Certain quadratic or cubic terms can be compensated by velocity differences.

## Abstract

We develop techniques to capture the effect of transport on the long-term dynamics of small, localized initial data in nonlinearly coupled reaction-diffusion-advection equations on the real line. It is well-known that quadratic or cubic nonlinearities in such systems can lead to growth of small, localized initial data and even finite time blow-up. We show that, if the components exhibit different velocities, then quadratic or cubic mix-terms, i.e. terms with nontrivial contributions from both components, are harmless. We establish global existence and diffusive Gaussian-like decay for exponentially and algebraically localized initial conditions allowing for quadratic and cubic mix-terms. Our proof relies on a nonlinear iteration scheme that employs pointwise estimates. The situation becomes very delicate if other quadratic or cubic terms are present in the system. We provide an example where a quadratic mix-term and a Burgers'-type coupling can compensate for a cubic term due to differences in velocities.