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MiS Preprint
50/2011
Stability of solutions to abstract evolution equations with delay
Alexander Ramm
Abstract
An equation $\dot{u}=A(t)u +B(t)F(t,u(t-\tau))$, $u(t)=v(t), -\tau \leq t \leq 0$ is considered, $A(t)$ and $B(t)$ are linear operators in a Hilbert space $H$, $\dot{u}=\frac{du}{dt}$, $F: H \to H$ is a non-linear operator, $\tau > 0$ is a constant. Under some assumption on $A(t), B(t)$ and $F(t,u)$ sufficient condittions are given for the solution $u(t)$ to exist globally, i.e, for all $t \geq 0$, to be globally bounded, and to tend to zero as $t \to \infty$.
