Global existence for quadratic systems of reaction-diffusion

We prove global existence in time of weak solutions to a class of quadratic reaction-diffusion systems for which a Lyapounov structure of LlogL-entropy type holds. The approach relies on an a priori dimension- independent L-2-estimate, valid for a wider class of systems including also some classical Lotka-Volterra systems, and which provides an L-1-bound on the nonlinearities, at least for not too degenerate diffusions. In the more degenerate case, some global existence may be stated with the use of a weaker notion of renormalized solution with defect measure, arising in the theory of kinetic equations.