Stability of Solutions to Evolution Problems

Alexander G. Ramm
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Alexander G. Ramm: Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA

Mathematics, 2013, vol. 1, issue 2, 1-19

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 ? : = d u d t , u = u ( t ) ? H , H is a Hilbert space, t ? R + : = [ 0 , ? ) , A ( t ) is a linear dissipative operator: Re ( A ( t ) u , u ) ? ? ? ( 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 ) ? ? ? ( t ) , and ? ( t ) ? 0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The non-classical case ? ( t ) ? 0 is also treated.

Keywords: Lyapunov stability; large-time behavior; dynamical systems; evolution problems; nonlinear inequality; differential equations (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2013
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