Asymptotic stability of solutions to abstract differential equations

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$(A(t)u,u)\leq -γ(t)\|u\|^2$ $\forall u\in H$, where $γ(t)>0$, and the case when $\lim_{t\to \infty}γ(t)=0$ 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.

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