  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.
