University of Heidelberg, Applied Analysis Seminar
Tue 05.03.2019
14:00, posted in Applied Analysis
It is well-known that quadratic or cubic nonlinearities in nonlinear 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 group 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 exponentially and polynomially localized initial data. Our approach can be related to the space-time resonances approach as developed by Germain, Masmoudi and Shatah in the dispersive setting. If nonlinear couplings which are not of mix-type are present in the system, the analysis breaks down as the spatial localization imposed by the Gaussian ansatz is too restrictive. For instance, the question whether a ‘seemingly harmless’ Burgers-type nonlinearity can be included in the system is rather subtle.

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