Existence and nonexistence results for higher-order semilinear evolution inequalities with critical potential

Abstract

We prove nonexistence results for higher-order semilinear evolution equations and inequalities of the form ∂ k u ∂ t k − Δ u + λ | x | 2 u ⩾ | u | q in R N × ( 0 , ∞ ) , where λ ⩾ − ( N − 2 2 ) 2 . This problem can be seen as a higher-order evolution version of the nonlinear Wheeler–De Witt equation which appears in the theory of quantum cosmology. In order to show that our result is sharp in the parabolic case, we establish the existence of positive solutions to the semilinear equation ∂ u ∂ t − Δ u + λ | x | 2 u ⩾ u q in R N × ( 0 , ∞ ) , for λ ⩾ 0 . The nonexistence results are based on the test function method, developed by Mitidieri, Pohozaev, Tesei and Véron. The existence result is established by the construction of an explicit global solution of the semilinear parabolic inequality.