We prove the existence of global strong solutions to the triangular Shigesada-Kawasaki-Teramoto (SKT) cross-diffusion system with Lokta-Volterra reaction terms in three dimensions.
A key part is the independent careful study of the parabolic equation \(a\partial_t w - \Delta w = f\) with a rough coefficient \(a\), homogeneous Neumann boundary conditions, and the special assumption \(\partial_tw \ge 0\).
By the same method, we obtain estimates for solutions to reaction-diffusion systems modelling reversible chemistry.