Global nonexistence for abstract evolution equations with positive initial energy
The authors consider the following initial value problem for abstract evolution equations \[ Pu_{tt}+Q(t)u_{t} +A(t,u)=F(t,u),\qquad t\in J=[0,\infty), \] with \(P\) and \(Q\) being linear self-adjoint operators, while \(A(t,u)\) and \(F(t,u)\) are, respectively, a linear operator in \(u\) and a nonlinear driving force. Introducing an appropriate coercive operator associated with the equation above, the authors prove noncontinuation (blow-up) of solutions when the initial energy is positive.
abstract second-order evolution equation
scientific article; zbMATH DE number 1241039