Abstract
We study for the first time the large-time behavior of solutions to certain hyperbolic inequalities with nonlinearities involving the function and its gradient. Namely, we consider the problem u(tt)-Delta u >=vertical bar u vertical bar(p)+vertical bar del u vertical bar(q) + f (t, x) in (0, oo) x R-N, where p, q > 1 and f >= 0, f not equivalent to 0. We obtain general criteria for the nonexistence of global solutions to this problem. Next, we discuss some special cases of the inhomogeneous term f. In particular, when N >= 3 and f depends only on the variable space, we obtain a discontinuous Fujita critical exponent p* (N, q). Namely, p* (N, q) = 1 + 2/N-2 if q > 1+ 1/N-1 , and p* (N,q) = infinity if q < 1 +1/N-1. Next, we extend our study to the exterior problem u(tt) - Delta u >= vertical bar u vertical bar(P) +vertical bar del u vertical bar(q) in (0, infinity) x D-c, under the boundary condition partial derivative u/partial derivative n+ >= f(t, x) in (0, infinity) x partial derivative D, where D is the unit open ball in R-N, D-c = R-N\D and n(+) is the outward (relative to D-c) unit normal of partial derivative D. (C) 2020 Elsevier Inc. All rights reserved.