Necessary and Sufficient Conditions for the Unique Solvability of a Nonlinear Reaction-Diffusion Model

It has been known for some time that a nonlinear reaction-diffusion model, with Dirichlet boundary conditions, is uniquely solvable if the reaction term satisfies an appropriate Lipschitz condition. However, as recently shown for an absorption model, such a condition is not necessary. We establish a uniqueness result which, in the case of reaction and diffusion governed by power laws, is in fact both necessary and sufficient for the unique solvability of the model. The improvement that is needed on the above-mentioned Lipschitz condition occurs in the so-called fast diffusion model