Stability of solutions to some evolution problem

Details Stability of solutions to some evolution problem Stability of solutions to some evolution problem

2010-12-13

arxiv.org/abs/1012.2785v1

Mathematics

Dynamical Systems

Scientific paper

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: $$ \dot{g}(t)\leq -\gamma(t)g(t)+\alpha(t,g(t))+\beta(t),\ t\geq 0;\quad g(0)=g_0. $$

Affiliated with

Also associated with

No associations

Rating

Stability of solutions to some evolution problem does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Stability of solutions to some evolution problem, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Stability of solutions to some evolution problem will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-27584