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We develop techniques to capture the effect of transport on the long-termdynamics of small, localized initial data in nonlinearly coupledreaction-diffusion-advection equations on the real line. It is well-known thatquadratic 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 thecomponents exhibit different velocities, then quadratic or cubic mix-terms,i.e. terms with nontrivial contributions from both components, are harmless. Weestablish global existence and diffusive Gaussian-like decay for exponentiallyand algebraically localized initial conditions allowing for quadratic and cubicmix-terms. Our proof relies on a nonlinear iteration scheme that employspointwise estimates. The situation becomes very delicate if other quadratic orcubic terms are present in the system. We provide an example where a quadraticmix-term and a Burgers'-type coupling can compensate for a cubic term due todifferences in velocities.

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