Logistic diffusion equations governed by the superposition of operators of mixed fractional order

2025/01/22 by Serena Dipierro, Dipierro, Serena, Edoardo Proietti Lippi +5 · 4 citations
Mathematics · #Fractional Differential Equations Solutions #Differential Equations and Numerical Methods #Differential Equations and Boundary Problems

paper · pdf · doi:10.48550/arxiv.2501.12967

Abstract

We discuss the existence of stationary solutions for logistic diffusion equations of Fisher-Kolmogoroff-Petrovski-Piskunov type driven by the superposition of fractional operators in a bounded region with "hostile" environmental conditions, modeled by homogeneous external Dirichlet data. We provide a range of results on the existence and nonexistence of solutions tied to the spectral properties of the ambient space, corresponding to either survival or extinction of the population. We also discuss how the possible presence of nonlocal phenomena of concentration and diffusion affect the endurance or disappearance of the population. In particular, we give examples in which both classical and anomalous diffusion leads to the extinction of the species, while the presence of an arbitrarily small concentration pattern enables survival.

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• A logistic equation with non-local operators of order near zero
• On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''
• Nontrivial solutions for nonlinear problems driven by a superposition of fractional p-Laplacians with Neumann boundary conditions
• Superlinear problems involving nonlinear superposition operators of mixed fractional order

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