@MISC{Ramm13stabilityof, author = {Alexander G. Ramm}, title = { Stability of Solutions to Evolution Problems}, year = {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) = u0. (∗) Here ˙u: = du, u = u(t) ∈ H, H is a Hilbert space, t ∈ R+:= dt [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) ‖ ≤ c0‖u‖p, p> 1, c0 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.