Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 09, pp. 1–13. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu NONEXISTENCE OF GLOBAL SOLUTIONS TO THE SYSTEM OF SEMILINEAR PARABOLIC EQUATIONS WITH BIHARMONIC OPERATOR AND SINGULAR POTENTIAL SHIRMAYIL BAGIROV Communicated by Ludmila S. Pulkina Abstract. In the domain Q 0 R = {x : |x| >R}× (0, +∞) we consider the problem ∂u 1 ∂t +Δ 2 u 1 - C 1 |x| 4 u 1 = |x| σ 1 |u 2 | q 1 , u 1 | t=0 = u 10 (x) ≥ 0, ∂u 2 ∂t +Δ 2 u 2 - C 2 |x| 4 u 2 = |x| σ 2 |u 1 | q 2 , u 2 | t=0 = u 20 (x) ≥ 0, Z ∞ 0 Z ∂B R u i ds dt ≥ 0, Z ∞ 0 Z ∂B R Δu i ds dt ≤ 0, where σ i ∈ R, q i > 1, 0 ≤ C i < ( 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. 1. Introduction Let us introduce the following notation: x =(x 1 ,...,x n ) ∈ R n , n> 4, r = |x| = p x 2 1 + ··· + x 2 n , B R = {x; |x| <R}, B 0 R = {x; |x| >R}, B R1,R2 = {x; R 1 < |x| <R 2 }, Q R = B R × (0; +∞), Q 0 R = B 0 R × (0; +∞), ∂B R = {x; |x| = R}, ∇u = ( ∂u ∂x1 ,..., ∂u ∂xn ), C 4,1 x,t (Q 0 R ) is the set of functions that are four times continuously differentiable with respect to x and continuously differentiable with respect to t in Q 0 R . In the domain Q 0 R we consider the system of equations ∂u 1 ∂t +Δ 2 u 1 - C 1 |x| 4 u 1 = |x| σ1 |u 2 | q1 ∂u 2 ∂t +Δ 2 u 2 - C 2 |x| 4 u 2 = |x| σ2 |u 1 | q2 , (1.1) with the initial condition u i | t=0 = u i0 (x) ≥ 0, (1.2) 2010 Mathematics Subject Classification. 35A01, 35B33, 35K52, 35K91. Key words and phrases. System of semilinear parabolic equation; biharmonic operator; global solution; critical exponent; method of test functions. c 2018 Texas State University. Submitted November 12, 2017. Published January 6, 2018. 1