Global existence and decay in multi-component reaction-diffusion-advection systems with different velocities: oscillations in time and frequency

TL;DR

This paper introduces a new analytical method using Fourier space estimates and oscillation exploitation to prove global existence and decay in multi-component reaction-diffusion-advection systems with different velocities, accommodating complex nonlinearities.

Contribution

It presents an alternative approach employing $L^1$-$L^p$ estimates and oscillation analysis, extending results to systems with multiple components and nonlinearities of Burgers' type.

Findings

Global existence for small localized initial data

Decay rates established for multi-component systems

Method applicable to systems with cubic mixed-terms

Abstract

It is well-known that quadratic or cubic nonlinearities in reaction-diffusion-advection systems can lead to growth of solutions with small, localized initial data and even finite time blow-up. It was recently proved, however, that, if the components of two nonlinearly coupled reaction-diffusion-advection equations propagate with different velocities, then quadratic or cubic mixed-terms, i.e.~nonlinear terms with nontrivial contributions from both components, do not affect global existence and Gaussian decay of small, localized initial data. The proof relied on pointwise estimates to capture the difference in velocities. In this paper we present an alternative method, which is better applicable to multiple components. Our method involves a nonlinear iteration scheme that employs $L^1$-$L^p$ estimates in Fourier space and exploits oscillations in time and frequency, which arise due to differences in transport. Under the assumption that each component exhibits different velocities, we establish global existence and decay for small, algebraically localized initial data in multi-component reaction-diffusion-advection systems allowing for cubic mixed-terms and nonlinear terms of Burgers' type.