Asymptotic stability of solutions to abstract differential equations

Abstract

An evolution problem for abstract differential equations is studied. The typical problem is: $$\dot{u}=A(t)u+F(t,u),t\ge 0;u(0)=u_0;\dot{u}=\frac{du}{dt}(\ast )$$ Here $A(t)$ is a linear bounded operator in a Hilbert space $H$, and $F$ is a nonlinear operator, $\Vert F(t,u)\Vert \le c_0\Vert u{\Vert }^p,p>1$, $c_0,p=const>0$. It is assumed that Re$(A(t)u,u)\le -\gamma (t)\Vert u{\Vert }^2$ $\forall u\in H$, 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.