A nonexistence result for a nonlinear wave equation with damping on a Riemannian manifold

Abstract

In this paper, we study the global nonexistence of solutions to a nonlinear wave equation with critical potential V(x) on a Riemannian manifold, the form of which is more general than those in (Todorova and Yordanov in C. R. Acad. Sci., Sér. 1 Math. 300: [557][558][559][560][561][562] 2000). The way we follow is motivated by the work of Qi S. Zhang (C. R. Acad. Sci., Sér. 1 Math. 333:109-114, 2001). We also prove the local existence and uniqueness result.

Nonexistence for hyperbolic problems on Riemannian manifolds1

“…In [16] some nonexistence results for problem (1.3) have been stated; however, in the proofs in [16] it is implicitly assumed that u ≥ 0 in M × (0, ∞) and that x → r(x) is of class C 2 in M \ {o}. Observe that the hypothesis that u ≥ 0 in M × (0, ∞) is not natural for solutions of hyperbolic equations; in fact, also in [13], where M = R n , sign-changing solutions were considered.…”