A semigroup approach to a reaction–diffusion system with cross-diffusion

Abstract

In this paper, we prove existence, uniqueness and special properties of the solution to a parabolic–elliptic reaction–diffusion system with cross-diffusion in an open bounded domain in Rd, as well as in Rd,d=1, 3, for various classes of the system parameters. In the case of a bounded domain and of Neumann boundary conditions we develop an approach based on a semigroup technique in the dual space (H1(Ω))′. This gives the possibility of considering singular initial data, distributions or functionals that may represent relevant physical data. The influence of the system parameters upon the qualitative behavior of the solution is investigated and some sufficient conditions for the existence of global or local solutions, and for global solutions in the case of small initial data are provided. In the case of the whole space, the problem is studied for a singular diffusion expressed by a multivalued operator. Existence is proved by passing to the limit in an approximating problem for which existence follows via a semigroup approach in H−1(Rd), d=1, 3. An argument for the uniqueness is developed in Beppo Levi spaces. Finally, the results are illustrated by a model exhibiting a self-organized criticality.