Positive solutions of indefinite logistic growth models with flux-saturated diffusion
| Autor: | Pierpaolo Omari, Elisa Sovrano |
|---|---|
| Přispěvatelé: | Omari, Pierpaolo, Sovrano, Elisa |
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Boundary (topology)
01 natural sciences Elliptic boundary value problem mean curvature operator strong solution flux-saturated diffusion logistic-type equation 0101 mathematics indefinite weight Mathematics Mathematical physics Dirichlet problem bounded variation solution Mean curvature Continuous function (set theory) Applied Mathematics Bounded variation solution Flux-saturated diffusion Indefinite weight Logistic-type equation Mean curvature operator Positive solution Strong solution 010102 general mathematics Zero (complex analysis) positive solution 010101 applied mathematics Bounded function Constant (mathematics) Analysis |
| Popis: | This paper analyzes the quasilinear elliptic boundary value problem driven by the mean curvature operator − div ∇ u ∕ 1 + | ∇ u | 2 = λ a ( x ) f ( u ) in Ω , u = 0 on ∂ Ω , with the aim of understanding the effects of a flux-saturated diffusion in logistic growth models featuring spatial heterogeneities. Here, Ω is a bounded domain in R N with a regular boundary ∂ Ω , λ > 0 represents a diffusivity parameter, a is a continuous weight which may change sign in Ω , and f : [ 0 , L ] → R , with L > 0 a given constant, is a continuous function satisfying f ( 0 ) = f ( L ) = 0 and f ( s ) > 0 for every s ∈ ] 0 , L [ . Depending on the behavior of f at zero, three qualitatively different bifurcation diagrams appear by varying λ . Typically, the solutions we find are regular as long as λ is small, while as a consequence of the saturation of the flux they may develop singularities when λ becomes larger. A rather unexpected multiplicity phenomenon is also detected, even for the simplest logistic model, f ( s ) = s ( L − s ) and a ≡ 1 , having no similarity with the case of linear diffusion based on the Fick–Fourier’s law. |
| Databáze: | OpenAIRE |
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