A surprising regularizing effect of the nonlinear semigroup associated to the semilinear heat equation and applications to reaction diffusion systems

Abstract: In this paper we prove that positive weak solutions for quasilinear parabolic equations on bounded domains subject to homogenous Neumann boundary conditions becme classical and global under the unique condition that the reaction doesn't change sign after certain positive time. We apply this result to reaction diffusion systems and prove global existence of theirs positive weak solutions under the same condition on theirs reactions. The nonlinearities growth isn't taken in consideration. The proof is based on the maximum principle.