Non-trivial non-negative periodic solutions of a system of doubly degenerate parabolic equations with nonlocal terms

Abstract

The aim of the paper is to provide conditions ensuring the ex- istence of non-trivial non-negative periodic solutions to a system of doubly degenerate parabolic equations containing delayed nonlocal terms and satis- fying Dirichlet boundary conditions. The employed approach is based on the theory of the Leray-Schauder topological degree theory, thus a crucial purpose of the paper is to obtain a priori bounds in a convenient functional space, here L 2(QT ), on the solutions of certain homotopies. This is achieved under different assumptions on the sign of the kernels of the nonlocal terms. The considered system is a possible model of the interactions between two biologi- cal species sharing the same territory where such interactions are modeled by the kernels of the nonlocal terms. To this regard the obtained results can be viewed as coexistence results of the two biological populations under different intra and inter specific interferences on their natural growth rates

Nontrivial, nonnegative periodic solutions of a system of singular-degenerate parabolic equations with nonlocal terms

“…In the very recent paper [60], the authors replace the nonlocal terms of (1.1) by Ω K i (ξ, t)u(ξ, t)dξ, for i = 1, 3, and Ω K i (ξ, t)v(ξ, t)dξ, for i = 2, 4. By means of local conditions, different from those proposed in [25,26], the authors obtain the coexistence of the two species via a similar topological approach when p, q ≥ 2, m, n ≥ 1 and, thus, only the slow and normal diffusion occurs, i.e. m(p − 1) ≥ 1, n(q − 1) ≥ 1.…”

“…The aim of this paper is to extend the results of [25] and [26], concerning the existence of non-negative, non-trivial periodic solutions, to a system of singular-degenerate parabolic equations. To the best of our knowledge, this is the first result for the case when 1 < p, q < 2, m > p and n > q, also in the case of a single equation.…”

“…m(p − 1) ≥ 1, n(q − 1) ≥ 1. More precisely, models for the interaction between two biological species sharing the same isolated territory, with the interactions represented by means of the kernels K i , i = 1, 2, 3, 4, were considered in related systems of doubly degenerate parabolic equations in [25], [60] and in systems of porous medium equations in [26]. On the other hand, some previous biological models found in the literature, see e.g.…”

“…[1], [2], [51], [52] involve the p-Laplacian with p > 1 (and m = 1). Furthermore, we observe that the equations of the system we consider treat all the possible types of diffusion: slow, normal and fast, while in [25], [60] and [26] only the slow and normal diffusions were presented. In fact, as it can be easily checked, if p ∈ 1,…”

“…To the best of our knowledge, this is the first result for the case when 1 < p, q < 2, m > p and n > q, also in the case of a single equation. We recall that the cases p, q > 2, m, n > 1 and p = q = 2, m, n > 1 were treated respectively in [25] and [26], see also [21] and [22] for a system of anisotropic (p(x), q(x))-Laplacian parabolic equations, with p(x), q(x) > 2 in Ω, and m = n = 1. In the very recent paper [60], the authors replace the nonlocal terms of (1.1) by Ω K i (ξ, t)u(ξ, t)dξ, for i = 1, 3, and Ω K i (ξ, t)v(ξ, t)dξ, for i = 2, 4.…”