In this paper, we obtain nonexistence results for global solutions to the system of higher-order semilinear partial differential inequalities displaylinesfracpartialkuiβpartialtkβDelta(aiβ(x,t)uiβ(x,t))geqtgammai+1ββ£xβ£sigmai+1ββ£ui+1β(x,t)β£pi+1β,quad1leqileqn,crun+1β=u1β, in cones and cone-like domains in mathbbRN, 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