Stability of solutions to abstract differential equations

TL;DR

This paper establishes a sufficient condition for the asymptotic stability of the zero solution in an abstract nonlinear evolution equation, even when the linear operator's spectrum approaches the imaginary axis over time.

Contribution

It provides a stability criterion for nonlinear evolution equations with time-varying operators whose spectra may approach the imaginary axis, extending previous spectral stability results.

Findings

Stability condition holds without fixed spectral bounds.

Spectrum of $A(t)$ can tend to the imaginary axis as $t\to\infty$.

Zero solution remains asymptotically stable under given conditions.

Abstract

A sufficient condition for asymptotic stability of the zero solution to an abstract nonlinear evolution problem is given. The governing equation is $\dot{u}=A(t)u+F(t,u),$ where $A(t)$ is a bounded linear operator in Hilbert space $H$ and $F(t,u)$ is a nonlinear operator, $\|F(t,u)\|\leq c_0\|u\|^{1+p}$, $p=const >0$, $c_0=const>0$. It is not assumed that the spectrum $\sigma:=\sigma(A(t))$ of $A(t)$ lies in the fixed halfplane Re$z\leq -\kappa$, where $\kappa>0$ does not depend on $t$. As $t\to \infty$ the spectrum of $A(t)$ is allowed to tend to the imaginary axis.