Nonexistence of global solutions to the system of semilinear parabolic equations with biharmonic operator and singular potential

Abstract

In the domain Q′R = {x : |x| > R} x (0, +∞) we consider the problem ∂u1/∂t + ∆2u1 - C1/|x|4 u1 = |x|σ1|u2|q1, u1|t=0 = u1 0(x) ≥ 0, ∂u2/∂t + ∆2u2 - C2/|x|4 u2 = |x|σ2|u1|q2, u2|t=0 = u2 0(x) ≥ 0, ∫∞0 ∫∂BR ui ds dt ≥ 0, ∫∞0 ∫∂BR ∆ui ds dt ≤ 0, where σi ∈ ℝ, qi > 1, 0 ≤ Ci < (n(n-4)/4)2, i = 1, 2. Sufficient condition for the nonexistence of global solutions is obtained. The proof is based on the method of test functions.