Stability of solutions to some abstract evolution equations with delay

TL;DR

This paper investigates the global existence, boundedness, and stability of solutions to a class of delay differential equations in Hilbert spaces, allowing for non-negative spectral bounds and nonlinear growth conditions.

Contribution

It provides new sufficient conditions ensuring global solutions and stability for abstract delay equations with non-negative spectral bounds and nonlinear terms.

Findings

Established conditions for global existence of solutions.

Proved solutions can converge to zero despite non-negative spectral bounds.

Extended stability analysis to nonlinear delay differential equations.

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 $\|G(t,u)\| \le \alpha(t)\|u\|^p$, $p>1$, $t\ge 0$. 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.