Stability of solutions to some evolution problem

Abstract

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