Mathematics 2013, 1(2), 4664; doi:10.3390/math1020046
Stability of Solutions to Evolution Problems
Mathematics Department, Kansas State University, Manhattan, KS 665062602, USA
Received: 26 February 2013 / Revised: 25 April 2013 / Accepted: 25 April 2013 / Published: 13 May 2013
View FullText

Download PDF [259 KB, uploaded 13 May 2013]
Abstract
Large time behavior of solutions to abstract differential equations is studied. The results give sufficient condition for the global existence of a solution to an abstract dynamical system (evolution problem), for this solution to be bounded, and for this solution to have a finite limit as $\text{t}\to \text{}\infty $ , in particular, sufficient conditions for this limit to be zero. The evolution problem is: $\dot{u}\text{}=\text{}A\left(t\right)u\text{}+\text{}F(t,\text{}u)\text{}+\text{}b\left(t\right),\text{}t\text{}\ge \text{}0;\text{}u\left(0\right)\text{}=\text{}{u}_{0}.$ (*) Here $\dot{u}\text{}:=\text{}\frac{du}{dt}\text{},\text{}u\text{}=\text{}u\left(t\right)\text{}\in \text{}H,\text{}H$ is a Hilbert space, $t\text{}\in \text{}{R}_{+}\text{}:=\text{}[0,\infty ),\text{}A\left(t\right)$ is a linear dissipative operator: $\text{Re}\left(A\right(t)u,u)\text{}\le \gamma \left(t\right)(u,\text{}u)$ where $F(t,\text{}u)$ is a nonlinear operator, $\Vert F(t,\text{}u)\text{}\Vert \text{}\le \text{}{c}_{0}{\Vert u\Vert}^{p},\text{}p\text{}>\text{}1,\text{}{c}_{0}$ and p are positive constants, $\Vert b\left(t\right)\text{}\Vert \text{}\le \text{}\beta \left(t\right)$, and $\beta \left(t\right)\ge 0$ is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The nonclassical case $\gamma \left(t\right)\text{}\le \text{}0$ is also treated. View FullTextKeywords:
Lyapunov stability; largetime behavior; dynamical systems; evolution problems; nonlinear inequality; differential equations
This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).
Scifeed alert for new publications
Never miss any articles matching your research from any publisher Get alerts for new papers matching your research
 Find out the new papers from selected authors
 Updated daily for 49'000+ journals and 6000+ publishers
 Define your Scifeed now
Related Articles
Article Metrics
Comments
[Return to top]
Mathematics
EISSN 22277390
Published by MDPI AG, Basel, Switzerland
RSS
EMail Table of Contents Alert