Stability of solutions to some evolution problem (1012.2785v1)

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