Semilinear hyperbolic inequalities with Hardy potential in a bounded domain

Abstract

We consider hyperbolic inequalities with Hardy potential u t t − Δ u + λ | x | 2 u ⩾ | x | − a | u | p in ( 0 , ∞ ) × B 1 ∖ { 0 } , u ( t , x ) ⩾ f ( x ) on ( 0 , ∞ ) × ∂ B 1 , where B 1 is the unit ball in R N , N ⩾ 3, λ > − ( N − 2 2 ) 2 , a ⩾ 0, p > 1 and f is a nontrivial L 1 -function. We study separately the cases: λ = 0, − ( N − 2 2 ) 2 < λ < 0 and λ > 0. For each case, we obtain an optimal criterium for the nonexistence of weak solutions. Our study yields naturally optimal nonexistence results for the corresponding stationary problem. The novelty of this work lies in two facts: (i) To the best of our knowledge, in all previous works dealing with nonexistence results for evolution equations with Hardy potential in a bounded domain, only the parabolic case has been investigated, making use of some comparison principles. (ii) To the best of our knowledge, in all previous works, the issue of nonexistence has been studied only in the case of positive solutions. In this paper, there is no restriction on the sign of solutions.