Abstract
This paper considers the properties of positive solutions for a nonlocal equation with nonlocal boundary condition on. The conditions on the existence and nonexistence of global positive solutions are given. Moreover, we establish the uniform blow-up estimates for the blow-up solution.
Highlights
In this paper, we consider the following nonlocal equation with nonlocal boundary condition: ut = Δu + uq(y, t)d y − kup, x ∈ Ω, t > 0, Ω u(x,t) = f (x, y)u(y,t)d y, x ∈ ∂Ω, t > 0, u(x, 0) = u0(x), x ∈ Ω, (1.1)where p, q ≥ 1, k > 0, and Ω ⊂ RN is a bounded domain with smooth boundary
Where p, q ≥ 1, k > 0, and Ω ⊂ RN is a bounded domain with smooth boundary
Many physical phenomena were formulated into nonlocal mathematical models and studied by many authors
Summary
We consider the following nonlocal equation with nonlocal boundary condition: ut = Δu + uq(y, t)d y − kup, x ∈ Ω, t > 0, Ω u(x,t) = f (x, y)u(y,t)d y, x ∈ ∂Ω, t > 0, u(x, 0) = u0(x), x ∈ Ω,. Where p, q ≥ 1, k > 0, and Ω ⊂ RN is a bounded domain with smooth boundary. The function f (x, y) ≡ 0 is nonnegative, continuous, and defined for x ∈ ∂Ω, y ∈ Ω, while u0 is a nonnegative continuous function and satisfies the compatibility condition u0(x) = Ω f (x, y)u0(y)d y for x ∈ ∂Ω. Many physical phenomena were formulated into nonlocal mathematical models (see [1,2,3]) and studied by many authors. In recent few years, the reaction-diffusion equation with nonlocal source has been studied extensively.
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