Global nonexistence for abstract evolution equation with positive initial energy

In this paper we treat a blow-up problem for nonlinear evolution equations of the second order in a general setting. Roughly speaking, the degrees of nonlinearity of A(t,u) and F(t,u) are assumed to be q−1 and p−1, respectively, with 1<q<p. By use of the so-called "concavity method'' due to H. Levine, we give a noncontinuation result for solutions whose initial energy is allowed to be positive. The result is a generalization of recent works by Levine and ourselves [in Harmonic analysis and nonlinear differential equations (Riverside, CA, 1995), 253-263, Contemp. Math., 208, Amer. Math. Soc., Providence, RI, 1997] and K. Ono [J. Differential Equations 137 (1997), 273-301].