Global existence for a class of quadratic reaction–diffusion systems with nonlinear diffusions and L1L1 initial data

In this work, existence of global weak solutions in any space dimension is proven for a class of reaction–diffusion systems with L1L1-initial data, nonlinear diffusions and at most quadratic reactions. The proof relies on a dimension-independent L2L2-estimate, based on a total mass control assumption. If initial data are in L2L2, this estimate provides a control of the quadratic nonlinearities in L1L1 up to t=0t=0. In the case of L1L1-initial data, we prove that the L2L2-estimate can be localized in time, which allows to pass to the limit in an approximate system for t>0t>0. We then prove the continuity of the solution in L1L1 at t=0t=0.