A study of pattern forming systems with a fully nonlocal interaction kernel of possibly changing sign

Abstract

A nonlocal model with a possibly sign changing kernel is proposed to study pattern formation problems in physical and biological systems with self-organization properties. One defines pre-solutions of a related linear problem and gives sufficient conditions for pre-solutions to be jump discontinuous solutions of the nonlocal model. Among a multitude of solutions a selection criterion is used to single out more significant critical solutions. These solutions are further classified into minimizing solutions, maximizing solutions and saddle solutions. Minimizing solutions are the most relevant for patterned states and they exist if the kernel changes sign. A non-negative kernel on the other hand behaves in some ways like the gradient term in the standard local model.