Global existence for reaction–diffusion systems with nonlinear diffusion and control of mass

Abstract

We prove here global existence in time of weak solutions for some reaction–diffusion systems with natural structure conditions on the nonlinear reactive terms which provide positivity of the solutions and uniform control of the total mass. The diffusion operators are nonlinear, in particular operators of the porous media type $u_i\mapsto -d_i\mathrm{\Delta }{u}_i^{m_i}$. Global existence is proved under the assumption that the reactive terms are bounded in $L^1$. This extends previous similar results obtained in the semilinear case when the diffusion operators are linear of type $u_i\mapsto -d_i\mathrm{\Delta }{u}_i$.