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Mathematics 2013, 1(2), 46-64; doi:10.3390/math1020046
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
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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 t , in particular, sufficient conditions for this limit to be zero. The evolution problem is: u ˙ = A(t)u + F(t, u) + b(t), t 0; u(0) = u 0 . (*) Here u ˙ := du dt , u = u(t) H, H is a Hilbert space, t R + := [0,), A(t) is a linear dissipative operator: Re(A(t)u,u) γ(t)(u, u) where F(t, u) is a nonlinear operator, F(t, u) c 0 u p , p > 1, c 0 and p are positive constants, b(t) β(t) , and β(t)0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The non-classical case γ(t) 0 is also treated.
Keywords: Lyapunov stability; large-time behavior; dynamical systems; evolution problems; nonlinear inequality; differential equations 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 3.0).

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

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