We consider an Klein–Gordon relativistic equation with a boundary dissipation of fractional derivative type. We study of stability of the system using semigroups theory and classical theorems over asymptotic behavior.
W.Arendt and C.J.K.Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc.306(2) (1988), 837–852. doi:10.1090/S0002-9947-1988-0933321-3.
2.
A.Borichev and Y.Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann.347(2) (2010), 455–478. doi:10.1007/s00208-009-0439-0.
3.
J.Choi and R.Maccamy, Fractional order Volterra equations with applications to elasticity, J. Math. Anal. Appl.139 (1989), 448–464. doi:10.1016/0022-247X(89)90120-0.
4.
A.Kilbas and J.Trujillo, Differential equation of fractional order: Methods, results and problems, I Appl. Anal.78(2) (2002), 435–493, II Appl. Anal., 81, 1-2, 2001, 153–192. doi:10.1080/00036810108840931.
5.
F.Mainardi, On signal velocity of anomalous dispersive waves, Il Nuovo Cimento. B74 (1983), 52–58. doi:10.1007/BF02721684.
6.
F.Mainardi, Linear dispersive waves with dissipation, in: Wave Phenomena: Modern Theory and Applications, C.Rogers and T.B.Moodie, eds, North-Holland, Amsterdam, 1984, pp. 307–4317. doi:10.1016/S0304-0208(08)71274-X.
7.
B.Mbodje, Wave energy decay under fractional derivative controls, IMA Journal of Mathematical Control and Information.23 (2006), 237–257. doi:10.1093/imamci/dni056.
8.
B.Mbodje and G.Montseny, Boundary fractional derivative control of the wave equation, IEEE Trans. Autom. Control.40 (1995), 368–382.
9.
J.Prüss, On the spectrum of C0-semigroup, Trans. Am. Math. Soc.248 (1984), 847–867. doi:10.2307/1999112.
10.
S.Samko, A.Kilbas and O.Marichev, Integral and Derivatives of Fractional Order, Gordon Breach, New York, 1993.
