Abstract
AbstractIn this paper, we study the stationary solutions of the Logistic equation$$\begin{array}{} \displaystyle u_t=\mathcal {D}[u]+\lambda u-[b(x)+\varepsilon]u^p \text{ in }{\it\Omega} \end{array}$$with Dirichlet boundary condition, here 𝓓 is a diffusion operator andε> 0,p> 1. The weight functionb(x) is nonnegative and vanishes in a smooth subdomainΩ0ofΩ. We investigate the asymptotic profiles of positive stationary solutions with the critical valueλwhenεis sufficiently small. We find that the profiles are different between nonlocal and classical diffusion equations.
Highlights
Introduction and main resultsWhere Ω is a bounded smooth domain in RN (N ≥ ), λ and p > are constant, the coe cient b ∈ C(Ω ) is a nonnegative function
Introduction and main resultsIn this paper, we study the semilinear equation ut = D[u] + λu − b(x)up in Ω, (1.1)where Ω is a bounded smooth domain in RN (N ≥ ), λ and p > are constant, the coe cient b ∈ C(Ω ) is a nonnegative function
We study the semilinear equation ut = D[u] + λu − b(x)up in Ω, (1.1)
Summary
Where Ω is a bounded smooth domain in RN (N ≥ ), λ and p > are constant, the coe cient b ∈ C(Ω ) is a nonnegative function. If D[u] = ∆u, we know that (1.1) is the classical reaction-di usion equation, which is related to some curvature problems in Riemannian geometry and di erent di usion problem in physic and population dynamics [1, 2]. D[u] = J * u(x, t) − u(x, t) = J(x − y)u(y, t)dy − u(x, t), RN (1.1) is the nonlocal dispersal equation with nonnegative and continuous kernel function J(x), see [4, 5]. It is known that both the classical di usion equation and nonlocal dispersal equation are widely used to model di erent di usion phenomena from applications as well as pure mathematics, see for example [10,11,12,13].
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