Open AccessThis article is

- freely available
- re-usable

Article

# Stability of Solutions to Evolution Problems

Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA

Received: 26 February 2013 / Revised: 25 April 2013 / Accepted: 25 April 2013 / Published: 13 May 2013

Download PDF
[259 KB, 13 May 2013;
original version 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 non-classical case $\gamma \left(t\right)\text{}\le \text{}0$ is also treated.

*Keywords:*Lyapunov stability; large-time behavior; dynamical systems; evolution problems; nonlinear inequality; differential equations

*This is an open access article distributed under the Creative Commons Attribution License (CC BY) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

# Share & Cite This Article

**MDPI and ACS Style**

Ramm, A.G. Stability of Solutions to Evolution Problems. *Mathematics* **2013**, *1*, 46-64.

# Related Articles

# Article Metrics

# Comments

# Cited By

[Return to top]*Mathematics*EISSN 2227-7390 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert