Decay of solutions of a second order differential equation with non-smooth second member

Consider the following abstract initial value problem(⁎)|u″(t)+μ(t)Au(t)+a(|A−θ2u(t)|2)u(t)+b(|A−ηu′(t)|)Au′(t)=f(t)in (0,∞);u(0)=u0,u′(0)=u1 in a real separable Hilbert space H with norm |u|. Here A is a positive self-adjoint operator of H; μ(t),a(s),b(s) positive functions, f(t) a vectorial non-sm...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2015-03, Vol.423 (2), p.975-993
Main Authors: Miranda, M. Milla, Lourêdo, A.T., Medeiros, L.A.
Format: Article
Language:English
Subjects:
Decay of solution
Hyperbolic problem with a non-smooth second member
Second order differential equation
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Summary:Consider the following abstract initial value problem(⁎)|u″(t)+μ(t)Au(t)+a(|A−θ2u(t)|2)u(t)+b(|A−ηu′(t)|)Au′(t)=f(t)in (0,∞);u(0)=u0,u′(0)=u1 in a real separable Hilbert space H with norm |u|. Here A is a positive self-adjoint operator of H; μ(t),a(s),b(s) positive functions, f(t) a vectorial non-smooth function and θ, η real numbers. In this paper we study the existence, uniqueness and decay of solutions of problem (⁎). In our approach, we use the Theory of Self-Adjoint Operators in Hilbert spaces, the compactness Aubin–Lions Theorem and a Lyapunov functional.
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2014.10.024