Global Nonexistence for Abstract Evolution Equations with Positive Initial Energy

In this paper we consider the problem of noncontinuation of solutions of the initial value problem for abstract evolution equations of the formPutt+Q(t) ut+A(t, u)=F(t, u), t J=[0, ∞),wherePandQare linear self-adjoint operators, andA(t, u) andF(t, u) are respectively a linear operator inu(typically of differential type) and a nonlinear driving force. Our principal concern is the noncontinuation (or blow-up) of solutions when the initial energy is positive, but appropriately bounded.