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

TL;DR

This paper investigates the existence of solutions for logistic diffusion equations involving superpositions of fractional operators, highlighting how nonlocal effects influence population survival or extinction in bounded environments.

Contribution

It provides new results on existence and nonexistence of solutions for fractional diffusion equations with mixed operators, linking spectral properties to population outcomes.

Findings

Both classical and anomalous diffusion can cause extinction.

Small concentration patterns can enable survival.

Spectral properties determine solution existence.

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.