Weak solutions of the Shigesaka-Kawasaki-Teramoto equations and their attractors

TL;DR

This paper proves the global existence of weak solutions and attractors for the Shigesada-Kawasaki-Teramoto equations in dimensions up to 4, using finite difference methods and maximum principles.

Contribution

It establishes the first comprehensive proof of weak solutions and attractors for these equations under general coefficient conditions.

Findings

Existence of weak solutions in dimensions ≤ 4.

Construction of a weak global attractor.

Use of finite differences and Stampachia's maximum principle.

Abstract

We derive the global existence of weak solutions of the Shigesada-Kawasaki-Teramoto systems in space dimension less or equal to 4 with a rather general condition on the coefficients. The existence is established using finite differences in time with truncations and an argument of Stampachia's maximum principle to show the positivity of the solutions. We derive also the existence of a weak global attractor.