Local energy decay for the wave equation with a nonlinear time-dependent damping

References

• Lions , JL and Strauss , WA . 1965 . Some non-linear evolution equations . Bull SMF , 93 : 43 – 96 .
   Google Scholar
• Lax , PD and Phillips , RS . 1967 . “ Scattering theory ” . In Pure and Applied Mathematics , Vol. 26 , New York : Academics Press .
   Google Scholar
• Morawetz , CS , Ralston , JV and Strauss , WA . 1978 . A correction to: Decay of solutions of the wave equation outside nontrapping obstacles . Comm. Pure Appl. Math. , 31 ( 6 ) : 795 Decay of solutions of the wave equation outside nontrapping obstacles, Comm. Pure Appl. Math. 30(4) (1977), pp. 447–508
   Web of Science ®Google Scholar
• Ralston , J . 1969 . Solutions of the wave equation with localized energy . Commun. Pure Appl. Math. , 22 : 807 – 823 .
   Web of Science ®Google Scholar
• Melrose , R . 1979 . Singularites and energy decay in acoustical scattering . Duke Math. J , 46 : 43 – 59 .
   Web of Science ®Google Scholar
• Burq , N . 1998 . Décroissance de l'énergie locale de l'équation des ondes pour le problème exterieur . Acta Math , 180 : 1 – 29 .
   Web of Science ®Google Scholar
• Nakao , M . 1998 . Stabilization of local energy in an exterior domain for the wave equation with a localized dissipation . J. Differ. Eqns. , 148 : 388 – 406 .
   Web of Science ®Google Scholar
• Aloui , L and Khenissi , M . 2002 . Stabilisation de l'équation des ondes dans un domaine extérieur . Rev. Mat. Iberoamerica , 18 ( 1 ) : 1 – 16 .
   Google Scholar
• Khenissi , M . 2003 . Équation des ondes amorties dans un domaine extérieur . Bull. Soc. Math. Fr. , 131 ( 2 ) : 211 – 228 .
   Google Scholar
• Bardos , C , Lebeau , G and Rauch , J . 1992 . Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary . SIAM J. Control Optim. , 30 ( 5 ) : 1024 – 1065 .
   Web of Science ®Google Scholar
• Gérard , P . 1991 . Microlocal defect measures . Commum. Par. Differ. Eqns. , 16 : 1761 – 1794 .
   Web of Science ®Google Scholar
• Lebeau , G . Équations des ondes amorties . Algebraic Geometric Methods in Mathematical Physics , (Kaciveli, 1993), Math. Phys. Stud., Vol. 19, Kluwer Acad. Publ., Dordrecht, 1996, pp. 73–109
   Google Scholar
• Daoulatli , M . 2007 . Local energy decay for the nonlinear dissipative wave equation in an exterior domain . Port. Math. (N.S.) , 64 ( 1 ) : 39 – 65 .
   Google Scholar
• Bchatnia , A and Daoulatli , M . 2004 . Scattering and exponential decay of the local energy for the solutions of semilinear and subcritical wave equation outside convex obstacle . Math. Z. , 247 : 619 – 642 .
   Web of Science ®Google Scholar
• Daoulatli , M , Dehman , B and Khenissi , M . 2010 . Local energy decay for the elastic system with nonlinear damping in an exterior domain . SIAM J. Control Optim. , 48 ( 8 ) : 5254 – 5275 .
   Web of Science ®Google Scholar
• Bellassoued , M . 2008 . Energy decay for the elastic wave equation with a local time-dependent non-linear damping . Acta Math. Sin., Engl. Ser. , 24 ( 7 ) : 1175 – 1192 .
   Web of Science ®Google Scholar
• Daoulatli , M . 2011 . Rates of decay for the wave systems with time-dependent damping . Discrete Contin. Dyn. Syst. , 31 ( 2 ) : 407 – 443 .
   Web of Science ®Google Scholar
• Martinez , P . 2000 . Precise decay rate estimates for time-dependent dissipative systems . Isr. J. Math. , 119 : 291 – 324 .
   Web of Science ®Google Scholar
• Messaoudi , SA and Mustafa , MI . 2010 . General energy decay rates for a weakly damped wave equation . Commun. Anal. , 9 ( 2 ) : 67 – 76 .
   Google Scholar
• Nakao , M . 1997 . On the decay of solutions of the wave equation with a local time-dependent non-linear dissipation . Adv. Math. Sci. Appl. , 7 : 317 – 331 .
   Google Scholar
• Lasiecka , I and Toundykov , D . 2006 . Energy decay rates for the semilinear wave equation with nonlinear localized damping and source terms . Nonlinear Anal. , 64 : 1757 – 1797 .
   Web of Science ®Google Scholar
• Lasiecka , I and Tataru , D . 1993 . Uniform boundary stabilization of semi-linear wave equation with nonlinear boundary dissipation . Differ. Integ. Eqns. , 6 ( 3 ) : 507 – 533 .
   Google Scholar
• Daoulatli , M , Lasiecka , I and Toundykov , D . 2009 . Uniform energy decay for a wave equation with partially supported nonlinear boundary dissipation without growth restrictions . Discrete Contin. Dyn. Syst., Ser. S , 2 ( 1 ) : 67 – 94 .
   Web of Science ®Google Scholar