Global strong solutions for the triangular Shigesada-Kawasaki-Teramoto cross-diffusion system in three dimensions and parabolic regularisation for increasing functions

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 $a{\mathrm{\partial }}_{t}w-\mathrm{\Delta }w=f$ with a rough coefficient $a$, homogeneous Neumann boundary conditions, and the special assumption ${\mathrm{\partial }}_{t}w\ge 0$. By the same method, we obtain estimates for solutions to reaction-diffusion systems modelling reversible chemistry.