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 L log L-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.