Stability of Solutions to Evolution Problems

Alexander G. Ramm
Additional contact information
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 ) ≤ − 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 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2013
References: View complete reference list from CitEc
Citations:

Downloads: (external link)
https://www.mdpi.com/2227-7390/1/2/46/pdf (application/pdf)
https://www.mdpi.com/2227-7390/1/2/46/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:1:y:2013:i:2:p:46-64:d:25671

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().