The diffusion equation with nonlocal data

A Nonlocal Convection-diffusion Equation

In this paper we study a nonlocal equation that takes into account convective and diffusive effects, ut = J ∗u−u+G ∗ (f(u))− f(u) in R, with J radially symmetric and G not necessarily symmetric. First, we prove existence, uniqueness and continuous dependence with respect to the initial condition of solutions. This problem is the nonlocal analogous to the usual local convection-diffusion equatio...

A Logistic Equation with Refuge and Nonlocal Diffusion

In this work we consider the nonlocal stationary nonlinear problem (J ∗ u)(x) − u(x) = −λu(x) + a(x)u(x) in a domain Ω, with the Dirichlet boundary condition u = 0 in R \ Ω and p > 1. The kernel J involved in the convolution (J ∗ u)(x) = RN 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 t...

Blowup Analysis for a Nonlocal Reaction Diffusion Equation with Potential

In this paper we investigate a nonlocal reaction diffusion equation with potential, under Neumann boundary. We obtain the complete classification of the parameters for which the solution blows up in finite time or exists globally. Moreover, we study the blowup rate and the blowup set for the blowup solution. Key–Words: Nonlocal diffusion, Blow up, Blowup rate, Blowup set

Quenching Time for a Nonlocal Diffusion Problem with Large Initial Data

Large Time Behavior for a Nonlocal Diffusion Equation with Absorption and Bounded Initial Data

Abstract. We study the large time behavior of nonnegative solutions of the Cauchy problem ut = ∫ J(x − y)(u(y, t) − u(x, t)) dy − u, u(x, 0) = u0(x) ∈ L , where |x|u0(x) → A > 0 as |x| → ∞. One of our main goals is the study of the critical case p = 1 + 2/α for 0 < α < N , left open in previous articles, for which we prove that t|u(x, t) − U(x, t)| → 0 where U is the solution of the heat equati...