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

Abstract

We study the existence of non-trivial, non-negative periodic solutions for systems of singulardegenerate parabolic equations with nonlocal terms and satisfying Dirichlet boundary conditions. The method employed in this paper is based on the Leray-Schauder topological degree theory. However, verifying the conditions under which such a theory applies is more involved due to the presence of the singularity. The system can be regarded as a possible model of the interactions of two biological species sharing the same isolated territory, and our results give conditions that ensure the coexistence of the two species.

Coexistence Solutions for a Periodic Competition Model with Singular–Degenerate Diffusion

“…The aim of this paper is to supplement the existing results with the case with gradient singularity. More specifically, concerning the existence of coexistence periodic solutions, the results in [12] together with our results will give a complete picture of the system (1.1)-(1.3) for all 1 < p, q < 2, m > 1, n > 1. In fact, the theory of topological degree has been used to deal with the case 1 < p, q < 2, m > p, n > q in [12], but it is difficult to apply this method to the general case due to the complicated, or even impossible, calculation of the topological degree.…”

“…More specifically, concerning the existence of coexistence periodic solutions, the results in [12] together with our results will give a complete picture of the system (1.1)-(1.3) for all 1 < p, q < 2, m > 1, n > 1. In fact, the theory of topological degree has been used to deal with the case 1 < p, q < 2, m > p, n > q in [12], but it is difficult to apply this method to the general case due to the complicated, or even impossible, calculation of the topological degree. Indeed, in order to obtain the topological degree of semi-non-trivial solutions such as (u, 0) and (0, v) to the approximate problem of (1.1)-(1.3), some estimates on the gradient of convenient powers of the solution are involved, and the technical restriction m > p, n > q is therefore imposed in [12] (see Proposition 2.3 and Lemma 2.4 therein).…”

“…Moreover, we are assuming that Ω is fully surrounded by a lethal environment, because both population densities are subject to homogeneous Dirichlet boundary conditions. For a detailed description of the model, we refer readers to [12] and the references therein.…”

“…For the case with gradient singularity, that is, 1 < p, q < 2, it was Fragnelli et al . who established the existence of coexistence periodic solutions to the problem (1.1)-(1.3) in the case of m > p, n > q, by applying the theory of Leray-Schauder degree (see [12]).…”

“…In fact, the theory of topological degree has been used to deal with the case 1 < p, q < 2, m > p, n > q in [12], but it is difficult to apply this method to the general case due to the complicated, or even impossible, calculation of the topological degree. Indeed, in order to obtain the topological degree of semi-non-trivial solutions such as (u, 0) and (0, v) to the approximate problem of (1.1)-(1.3), some estimates on the gradient of convenient powers of the solution are involved, and the technical restriction m > p, n > q is therefore imposed in [12] (see Proposition 2.3 and Lemma 2.4 therein). To overcome these difficulties, we apply the Leray-Schauder fixed-point theorem and propose a different approximate problem from which we are able to obtain the crucial a priori lower-bound estimates.…”