Large-time behavior of solutions to evolution problems

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 $\dot{u}:=\frac{du}{dt}$, $u=u(t)\in H$, $H$ is a Hilbert space, $t\in {\mathbb{R}}_{+}:=[0,\infty )$, $A(t)$ 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, $|F(t,u)|\le c_0|u{|}^p$, $p>1$, $c_0,p$ are positive 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 $t\ge 0$, to be bounded uniformly on ${\mathbb{R}}_{+}$, 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 $\beta (t)$.

arXiv admin note: substantial text overlap with arXiv:1012.2785