On blow-up set of solutions of initial boundary value problem for a system of parabolic equations with nonlocal boundary conditions

Abstract

We consider a system of semilinear parabolic equations u_t = Δг + с₁(x,t)v_p, v_t = Δv + c₂(x,t)u^q, (x,t) ∈ Ω × (0,+∞) with nonlinear nonlocal boundary conditions ∂u/∂η = ∫k₁(x,y,t)u^m(y,t)dy, ∂v/∂η = ∫_Ωk₂(x,y,t)v_n(y,t)dy, (x,t) ∈ ∂Ω × (0,+∞) and initial data u(x,0) = u₀(x), v(x,0) = v₀(x), x ∈ Ω, where p, q, m, n are positive constants, Ω is bounded domain in R^N(N ≥ 1) with a smooth boundary ∂Ω, η is unit outward normal on ∂Ω. Nonnegative locally Hӧlder continuous functions c_i(x,t),i = 1,2, are defined for x ∈ Ω, t ≥ 0; nonnegative continuous functions k_i(x,y,t),i = 1,2 are defined for x ∈ ∂Ω, y ∈ Ω, t ≥ 0; nonnegative continuous functions u₀(x),v₀(x) are defined for x ∈ Ω and satisfy the conditions ∂u₀(x)/∂η = ∫k₁(x,y,0)u^m₀(y)dy, ∂v₀(x)/∂η = ∫_Ωk₂(x,y,0)vⁿ₀(y)dy for x ∈ ∂Ω. In the paper blow-up set of classical solutions is investigated. It is established that blow-up of the solutions can occur only on the boundary ∂Ω if max(p,q) ≤ 1, max(m,n)> 1 and under certain conditions for the coefficients k_i(x,y,t),i = 1,2.