Stability of solutions to some abstract evolution equations with delay

N. S. Hoang, A. Ramm
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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.
一类具有时滞的抽象演化方程解的稳定性
研究了时滞微分方程(*)$\dot{u} = A(t)u + G(t,u(t-\tau)) + f(t)$, $t\ge 0$, $u(t) = v(t)$, $-\tau \le t\le 0$解的全局存在性和稳定性。这里$A(t):\mathcal{H}\to \mathcal{H}$是Hilbert空间$\mathcal{H}$中的一个封闭的、密集定义的线性算子,$G(t,u)$是$\mathcal{H}$中关于$u$和$t$的连续的非线性算子。我们假设$A(t)$的光谱位于半平面$\Re \lambda \le \gamma(t)$,其中$\gamma(t)$不一定是负的,并且$\|G(t,u)\| \le \alpha(t)\|u\|^p$, $p>1$, $t\ge 0$。在$\gamma(t)$可以取正值的非经典假设下,提出并证明了方程解全局存在、有界并在$t$趋于$\infty$时收敛于零的充分条件。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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