A logistic equation with refuge and nonlocal diffusion

By Jorge García-Melián, Julio D. Rossi. 2009

68 citations recorded by the source corpus.

In this work we consider the nonlocal stationary nonlinear problem $(J* u)(x) - u(x)= -\lambda u(x)+ a(x) u^p(x)$ in a domain $\Omega$, with the Dirichlet boundary condition $u(x)=0$ in $\mathbb{R}^N\setminus \Omega$ and $p>1$. The kernel $J$ involved in the convolution $(J*u) (x) = \int_{\mathbb{R}^N} J(x-y) u(y) dy$ is a smooth, compactly supported nonnegative function with unit integral, while the weight $a(x)$ is assumed to be nonnegative and is allowed to vanish in a smooth subdomain $\Omega_0$ of $\Omega$. Both when $a(x)$ is positive and when it vanishes in a subdomain, we completely discuss the issues of existence and uniqueness of positive solutions, as well as their behavior with respect to the parameter $\lambda$.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/f33e920b1b54b55adcd5d73560aa11c7bcc3c682

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In Applied Mathematics

In Computational Theory and Mathematics

Citations in the corpus

Pulsating fronts for nonlocal dispersion and KPP nonlinearity (2013, unknown): “1) we quote [1,2,18,20,21,23,25] for the existence and characterization of traveling fronts for this equation with homogeneous nonlinearity and [3,22,24,36,42] for the study of the stationary problem.”
On the definition and the properties of the principal eigenvalue of some nonlocal operators (2015, background): “When Ω is a bounded domain [3, 25, 28, 36, 41, 59], an optimal persistence criteria has been obtained using the sign of λp(MΩ + ∂uf(x, 0)), where MΩ stands for the operator: M Ω [φ] := ∫”