Stability of solutions to some evolution problem

Large time behavior of solutions to abstract differential equations is studied. The corresponding evolution problem is: $\dot{u}=A(t)u+F(t,u)+b(t), \quad t\ge 0; \quad u(0)=u_0. \qquad (*)$ Here $\dot{u}:=\frac {du}{dt}$, $u=u(t)\in H$, $t\in \R_+:=[0,\infty)$, $A(t)$ is a linear dissipative operator: Re$(A(t)u,u)\le -γ(t)(u,u)$, $γ(t)\ge 0$, $F(t,u)$ is a nonlinear operator, $\|F(t,u)\|\le c_0\|u\|^p$, $p>1$, $c_0,p$ are constants, $\|b(t)\|\le β(t),$ $β(t)\ge 0$ is a continuous function. Sufficient conditions are given for the solution $u(t)$ to problem (*) to exist for all $t\ge0$, to be bounded uniformly on $\R_+$, and a bound on $\|u(t)\|$ is given. This bound implies the relation $\lim_{t\to \infty}\|u(t)\|=0$ under suitable conditions on $γ(t)$ and $β(t)$. The basic technical tool in this work is the following nonlinear inequality: $\dot{g}(t)\leq -γ(t)g(t)+α(t,g(t))+β(t),\ t\geq 0;\quad g(0)=g_0.$

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