Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms
Genni Fragnelli, Paolo Nistri, Duccio Papini · Discrete and Continuous Dynamical Systems · 2011
The aim of the paper is to provide conditions ensuring theexistence of non-trivial non-negative periodic solutions to asystem of doubly degenerate parabolic equations containing delayednonlocal terms and satisfying Dirichlet boundary conditions. Theemployed approach is based on the theory of the Leray-Schaudertopological degree theory, thus a crucial purpose of the paper isto obtain a priori bounds in a convenient functional space, here$L^2(Q_T)$, on the solutions of certain homotopies. This isachieved under different assumptions on the sign of the kernels ofthe nonlocal terms. The considered system is a possible model ofthe interactions between two biological species sharing the sameterritory where such interactions are modeled by the kernels ofthe nonlocal terms. To this regard the obtained results can beviewed as coexistence results of the two biological populationsunder different intra and inter specific interferences on theirnatural growth rates.