Nonexistence criteria for systems of parabolic inequalities in an annulus

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We are concerned with nonexistence results for a class of systems of parabolic inequalities in (0,∞)×A, where A={x∈RN:1<|x|≤2}. The considered systems involve a singular potential function V(x)=(|x|−1)−ρ, ρ>0, in front of the power nonlinearities. Two types of inhomogeneous boundary conditions on ∂B2={x∈RN:|x|=2} are discussed: Neumann type conditions and Dirichlet type conditions. Using a unified approach, an optimal criterium of nonexistence is obtained for both cases. Our study yields naturally optimal nonexistence results for the corresponding stationary systems.