The effect of different velocities on global existence and decay in reaction-diffusion-advection systems
It is well-known that quadratic or cubic nonlinearities in reaction-diffusion systems can lead to growth of small initial data and even finite time blow-up. In this talk I will show that, if in nonlinearly coupled reaction-diffusion systems components exhibit different velocities, then quadratic or cubic mix-terms are harmless. Using a careful spatio-temporal analysis one can establish global existence and diffusive Gaussian-like decay for small, exponentially and polynomially localized initial data.
Our proof relies on a nonlinear iteration scheme that employs pointwise estimates. I will also present an alternative approach that can be related to the space-time resonances approach as developed by Germain, Masmoudi and Shatah in the dispersive setting.