ISSN 0012-2661, Differential Equations, 2016, Vol. 52, No. 4, pp. 467–482. c  Pleiades Publishing, Ltd., 2016. Original Russian Text c  A.L. Gladkov, A.I. Nikitin, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 4, pp. 490–505. PARTIAL DIFFERENTIAL EQUATIONS On the Existence of Global Solutions of a System of Semilinear Parabolic Equations with Nonlinear Nonlocal Boundary Conditions A. L. Gladkov and A. I. Nikitin Belarusian State University, Minsk, Belarus Vitebsk State University, Vitebsk, Belarus e-mail: gladkoval@mail.ru, ip.alexnikitin@gmail.com Received February 23, 2015 Abstract—We establish conditions for the existence and nonexistence of global solutions of an initial–boundary value problem for a system of semilinear parabolic equations with nonlin- ear nonlocal boundary conditions. The results depend on the behavior of variable coefficients as t →∞. DOI: 10.1134/S0012266116040078 1. INTRODUCTION In the present paper, we study the following initial–boundary value problem for a system of semilinear parabolic equations with nonlocal boundary conditions: u t =Δu + c 1 (x, t)v p , v t =Δv + c 2 (x, t)u q , x ∈ Ω, t> 0, u(x, t)=  Ω k 1 (x,y,t)u m (y,t) dy, x ∈ ∂ Ω, t> 0, v(x, t)=  Ω k 2 (x,y,t)v n (y,t) dy, x ∈ ∂ Ω, t> 0, u(x, 0) = u 0 (x), v(x, 0) = v 0 (x), x ∈ Ω, (1) where p, q, m, and n are positive constants; Ω is a bounded domain in R N (N ≥ 1) with smooth boundary ∂ Ω; c 1 (x, t) and c 2 (x, t) are nonnegative locally H¨ older continuous functions defined for x ∈ Ω and t ≥ 0; k 1 (x,y,t) and k 2 (x,y,t) are nonnegative continuous functions defined for x ∈ ∂ Ω, y ∈ Ω, and t ≥ 0; and u 0 (x) and v 0 (x) are nonnegative continuous functions satisfying the boundary conditions for t = 0. Let us introduce the definition of upper and lower solutions of problem (1). Let Q T =Ω × (0,T ). Definition 1. A pair (u, v) of nonnegative functions is called a lower solution of problem (1) in Q T if u, v ∈ C 2,1 (Q T ) ∩ C ( Q T ) and u t ≤ Δu + c 1 (x, t)v p , v t ≤ Δv + c 2 (x, t)u q , x ∈ Ω, 0 <t<T, u(x, t) ≤  Ω k 1 (x,y,t)u m (y,t) dy, x ∈ ∂ Ω, 0 <t<T, v(x, t) ≤  Ω k 2 (x,y,t)v n (y,t) dy, x ∈ ∂ Ω, 0 <t<T, u(x, 0) ≤ u 0 (x), v(x, 0) ≤ v 0 (x), x ∈ Ω. (2) 467