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Stability of Solutions to Evolution Problems
Mathematics Department, Kansas State University, Manhattan, KS 66506-2602, USA
Received: 26 February 2013; in revised form: 25 April 2013 / Accepted: 25 April 2013 / Published: 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 t —> oo, in particular, sufficient conditions for this limit to be zero. The evolution problem is: it = A(t)u + F (t , u) + b(t) , t > 0; u(0) = uo. (*) Here U := 2, u = u(t) E H, H is a Hilbert space, t E ILF := [0, oo), A(t) is a linear dissipative operator: Re(A(t)u, u) < —7 (t)(u , u), where F(t, u) is a nonlinear operator, 11 F (t , u)11 < collul IP' P > 1, co and p are positive constants, Ilb(t) 11 < 13 (t) , and 13 (t) > 0 is a continuous function. The basic technical tool in this work are nonlinear differential inequalities. The non-classical case -y (t) < 0 is also treated.
Keywords: Lyapunov stability; large-time behavior; dynamical systems; evolution problems; nonlinear inequality; differential equations
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MDPI and ACS Style
Ramm, A.G. Stability of Solutions to Evolution Problems. Mathematics 2013, 1, 46-64.
Ramm AG. Stability of Solutions to Evolution Problems. Mathematics. 2013; 1(2):46-64.
Ramm, Alexander G. 2013. "Stability of Solutions to Evolution Problems." Mathematics 1, no. 2: 46-64.