Asymptotic stability of solutions to abstract differential equations

TL;DR

This paper investigates the asymptotic stability of solutions to a class of abstract differential equations with time-dependent linear operators and nonlinear terms, providing decay rate estimates even when stability conditions weaken over time.

Contribution

It introduces new stability results for differential equations with time-varying operators and nonlinearities, including cases where the stability parameter tends to zero.

Findings

Established decay rate estimates for solutions.

Extended stability analysis to cases with diminishing stability parameters.

Provided conditions ensuring asymptotic stability.

Abstract

An evolution problem for abstract differential equations is studied. The typical problem is: $$\dot{u}=A(t)u+F(t,u), \quad t\geq 0; \,\, u(0)=u_0;\quad \dot{u}=\frac {du}{dt}\qquad (*)$$ Here $A(t)$ is a linear bounded operator in a Hilbert space $H$, and $F$ is a nonlinear operator, $\|F(t,u)\|\leq c_0\|u\|^p,\,\,p>1$, $c_0, p=const>0$. It is assumed that Re$(A(t)u,u)\leq -\gamma(t)\|u\|^2$ $\forall u\in H$, where $\gamma(t)>0$, and the case when $\lim_{t\to \infty}\gamma(t)=0$ is also considered. An estimate of the rate of decay of solutions to problem (*) is given. The derivation of this estimate uses a nonlinear differential inequality.