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# Stability of Solutions to Evolution Problems

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

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# 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 which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

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**MDPI and ACS Style**

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