Nonexistence of Solutions to Systems of Higher-order Semilinear Inequalities in Cone-like Domains

Abstract

In this paper, we obtain nonexistence results for global solutions to the system of higher-order semilinear partial differential inequalities ∂kui/ ∂tk - ∆(αi(x, t)ui(x, t)) ≥ tγi+1|x|σi+1|ui+1 (x, t)|Pi + 1, 1 ≤ i ≤ n, un+1 = u1, in cones and cone-like domains in ℝN, t > 0. Our results apply to nonnegative solutions and to solutions which change sign. Moreover, we provide a general formula of the critical exponent corresponding to this system. Our proofs are based on the test function method, developed by Mitidieri and Pohozaev.