Large time behavior of solutions to a class of hyperbolic inequalities with mixed nonlinearities

Abstract

We investigate the large time behavior of solutions to a class of inhomogeneous hyperbolic inequalities involving combined nonlinearities of the form |u|p+1Γ(σ)∫0t(t−s)σ−1|∇u(s,x)|qds+w(x), where p,q>1 and σ>0. We show that, if w has a positive average, then the considered class admits no global weak solution. Our approach is based on nonlinear capacity estimates specifically adapted to the nonlocal setting of our problem.