Global nonexistence for abstract evolution equations with positive initial energy

Abstract. In this paper we consider the problem of non-continuation of solutions of the initial value problem for abstract evolution equations of the form Putt +Q(t)ut +A(t, u) = F (t, u), t ∈ J = [0,∞) where P = and Q are linear self-adjoint operators, and A(t, u) and F (t, u) are respectively a linear operator in u (typically of differential type) and a nonlinear driving force. Our principal concern is the non-continuation (or blow–up) of solutions when the initial energy is positive, but appropriately bounded. §1. Introduction. evolution equations of the type (1.1) Putt +Q(t)ut +A(t, u) = F (t, u), t ∈ J = [0,∞), where P and Q(t) are linear self-adjoint operators, and A(t, u) and F (t, u) are typically