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

• Main Authors: Alexander L. Gladkov, Alexandr I. Nikitin
• Format: Article
• Language: Belarusian
• Published: Belarusian State University 2019-01-01
• Series: Журнал Белорусского государственного университета: Математика, информатика
• Subjects: system of semilinear parabolic equations; nonlocal boundary conditions; blow-up set
• Online Access: https://journals.bsu.by/index.php/mathematics/article/view/780

We consider a system of semilinear parabolic equations ut = Δг + с1(x,t)vp, vt = Δv + c2(x,t)uq, (x,t) ∈ Ω × (0,+∞) with nonlinear nonlocal boundary conditions ∂u/∂η = ∫k1(x,y,t)um(y,t)dy, ∂v/∂η = ∫Ωk2(x,y,t)vn(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 RN(N ≥ 1) with a smooth boundary ∂Ω, η is unit outward normal on ∂Ω. Nonnegative locally Hӧlder continuous functions ci(x,t),i = 1,2, are defined for x ∈ Ω, t ≥ 0; nonnegative continuous functions ki(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)um0(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 ki(x,y,t),i = 1,2.