On the initial-boundary value problem for a nonlocal parabolic equation with nonlocal boundary condition

Abstract

We consider a nonlinear nonlocal parabolic equation u_t = Δu + a(x,t)u_r∫_Ωu^p(y,t)dy - b(x,t)u^q for (x,t) ∈ Ω × (0,+∞) with nonlinear nonlocal boundary condition u(x,t)|∂_{Ω × (0,+∞)} = ∫_Ωk(x,y,t)u^l(y,t)dy and initial data u(x,0) = u₀(x), x ∈ Ω, where r, p, q, l are positive constants; Ω is a bounded domain in Rⁿ with smooth boundary ∂Ω. Nonnegative functions a(x,t) and b(x,t) are defined for x ∈ Ω, t ≥ 0 and local Hӧlder continuous, nonnegative continuous function k(x,y,t) is defined for x ∈ ∂Ω, y ∈ Ω, t ≥ 0, nonnegative continuous function u₀(x) is defined for x ∈ Ω and satisfies the condition u₀(x) = ∫_Ωk(x,y,0)u₀^t(y)dy for x ∈ ∂Ω. In this paper we study classical solutions. To prove the existence of a local maximal solution, we consider the regularization of the original problem. We establish the existence of a local solution of the regularized problem and the convergence of solutions of this problem to a local maximal solution of the original problem. We introduce definitions of a supersolution and a subsolution. It is shown that a supersolution is not less than a subsolution. We establish the positiveness of solutions of the problem with nontrivial initial data under certain conditions on the data of the problem. As a consequence of the positiveness of solutions and the comparison principle of solutions, we prove the uniqueness theorem.