Logistic Neumann problems with discontinuous coefficients
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stic Neumann problems with discontinuous coefficients Kazuaki Taira1 Received: 10 July 2020 / Accepted: 1 September 2020 / Published online: 18 September 2020 © Università degli Studi di Ferrara 2020
Abstract The purpose of this paper is for the first time to provide a careful and accessible exposition of the study of the existence of positive solutions of semilinear Neumann problems for diffusive logistic equations with discontinuous coefficients, which models population dynamics in environments with spatial heterogeneity. A biological interpretation of our main result is that when the environment has an impassable boundary and is on the average unfavorable, then high diffusion rates have the same effect (that is, the ultimate extinction of the population) as they always have when the boundary is deadly; but if the boundary is impassable and the environment is on the average neutral or favorable, then the population can persist, no matter what its rate of diffusion. The approach here is based on explicit representation formulas for the solutions of the Neumann problem and also on the L p boundedness of Calderón–Zygmund singular integral operators appearing in those representation formulas. That is why we consider the case where the space dimension is greater than 2. Moreover, we make use of an L p variant of an estimate for the Green operator of the Neumann problem introduced in the study of Feller semigroups. Keywords Neumann problem · Diffusive logistic equation · VMO function · Indefinite weight function · Singular integral · Representation formula · Bifurcation theory · The sub-super-solution method · Positive solution Mathematics Subject Classification 92D25 · 35R05 · 42B20 · 60J70
Contents 1 Introduction and main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 410 1.1 Neumann eigenvalue problems with indefinite weights . . . . . . . . . . . . . . . . . . . . 412
Dedicated to Professor Hiroki Tanabe on the occasion of his 90th birthday.
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Kazuaki Taira [email protected] Institute of Mathematics, University of Tsukuba, Tsukuba 305-8571, Japan
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ANNALI DELL’UNIVERSITA’ DI FERRARA (2020) 66:409–485
1.2 Logistic Neumann problems . . . . . . . . . . . . . . . . . . . . 1.3 An outline of the paper . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The spaces BMO and VMO . . . . . . . . . . . . . . . . . . . . . 2.2 The Kre˘ın and Rutman theorem . . . . . . . . . . . . . . . . . . . 2.3 Local bifurcation theory . . . . . . . . . . . . . . . . . . . . . . . 3 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The Neumann problem . . . . . . . . . . . . . . . . . . . . . . . 3.2 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proof of Theorem 1.1 . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Proof of Proposition 1.1 . . . . . . . . . . . . . . . . . . . . . . . 4 Proof of parts (i) and (ii) of Theorem 1.2 . . . . . . . . . . . . . .
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