Stability of solutions to some abstract evolution equations with delay

Abstract

The global existence and stability of the solution to the delay differential equation (*)$\dot{u}=A(t)u+G(t,u(t-\tau ))+f(t)$, $t\ge 0$, $u(t)=v(t)$, $-\tau \le t\le 0$, are studied. Here $A(t):\mathcal{H}\to \mathcal{H}$ is a closed, densely defined, linear operator in a Hilbert space $\mathcal{H}$ and $G(t,u)$ is a nonlinear operator in $\mathcal{H}$ continuous with respect to $u$ and $t$. We assume that the spectrum of $A(t)$ lies in the half-plane $\Re \lambda \le \gamma (t)$, where $\gamma (t)$ is not necessarily negative and $\Vert G(t,u)\Vert \le \alpha (t)\Vert u{\Vert }^p$, $p>1$, $t\ge 0$. Sufficient conditions for the solution to the equation to exist globally, to be bounded and to converge to zero as $t$ tends to $\infty$, under the non-classical assumption that $\gamma (t)$ can take positive values, are proposed and justified.

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