For S a positive selfadjoint operator on a Hilbert space, d2u dt (t)+2F(S)du dt (t)+S2u(t)=0 describes a class of wave equations with strong friction or damping if F is a positive Borel function. Under suitable hypotheses, it is shown that u(t)=v(t)+w(t) where v satisfies and 2F(S) dv dt (t)+S2v(t)=0 w(t) v(t) →0, as t →+∞. The required initial condition v(0) is given in a canonical way in terms of u(0), u′(0).
Fragnelli, G., Goldstein, G.R., Goldstein, J.A., Romanelli, S. (2013). Asymptotic parabolicity for strongly damped wave equations. In Spectral Analysis, Differential Equations and Mathematical Physics: A Festschrift in Honor of Fritz Gesztesy’s 60th Birthday (pp.119-131). American Mathematical Society [10.1090/pspum/087/01432].
Asymptotic parabolicity for strongly damped wave equations
G. FRAGNELLI
;
2013-01-01
Abstract
For S a positive selfadjoint operator on a Hilbert space, d2u dt (t)+2F(S)du dt (t)+S2u(t)=0 describes a class of wave equations with strong friction or damping if F is a positive Borel function. Under suitable hypotheses, it is shown that u(t)=v(t)+w(t) where v satisfies and 2F(S) dv dt (t)+S2v(t)=0 w(t) v(t) →0, as t →+∞. The required initial condition v(0) is given in a canonical way in terms of u(0), u′(0).
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https://hdl.handle.net/11365/1276637