Asymptotic stability of solutions to abstract differential equations

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.