Some new results on global nonexistence for abstract evolution equation with positive initial energy

Let V,Y be Hilbert spaces, and W,X be Banach spaces such that their intersection contains a nontrivial subspace G. Consider the linear operators P:V→V′, Q:J×Y→Y′, and the nonlinear mappings A:J×W→W′ and F:J×X→X′, where the prime designates a dual space and J=[0,∞). The operators P and Q(t) (t∈J) are assumed to be symmetric and nonnegative-definite, while A and F can be represented as the Fréchet derivatives of some C1 potential functions. We prove the nonexistence of global solutions u:J→G to the abstract equation Putt+Q(t)ut+A(t,u)=F(t,u), t∈J, under an appropriate restriction on the initial total energy of u. The result is applied to a partial differential equation of divergence form on a (possibly unbounded) domain in Rn. The paper generalizes earlier related work by H. A. Levine and ourselves [in Harmonic analysis and nonlinear differential equations (Riverside, CA, 1995), 253-263, Contemp. Math., 208, Amer. Math. Soc., Providence, RI, 1997], as well as a recent study by K. Ono [J. Differential Equations 137 (1997), 273-301].