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In this paper we prove the global existence and study the asymptotic behaviour of solutions to a degenerate wave equation with a nonlinear dissipative term.
[1]Arosio, A. and Gravaldi, S., ‘On the mildly degenerate Kirchhoff string’, Math. Methods Appl. Sci.14 (1991), 177–195.CrossRefGoogle Scholar
[2]
[2]Arosio, A. and Spagnolo, S., ‘Global solutions to the Cauchy problem for a nonlinear hyperbolic equation’, in Nonlinear PDE and their applications, (Brezis, H. and Lions, J. L., Editors), Collège de France seminar 6 (Pitman, Boston, 1984), pp. 1–26.Google Scholar
[3]
[3]Bernstein, B., ‘Sur une classe d'équations fonctionnelles aux dérivées partielles’, Izv. Akad. USSR Ser. Mat.4 (1940), 17–26.Google Scholar
[4]
[4]Crippa, H.R., ‘On local solutions of some mildly degenerate hyperbolic equations’, Nonlinear Anal.21 (1993), 565–574.CrossRefGoogle Scholar
[5]
[5]D'Ancona, P. and Shibata, Y., ‘On global solvability of non-linear viscoelastic equations in the analytic category’, Math. Methods Appl. Sci.17 (1994), 477–489.CrossRefGoogle Scholar
[6]
[6]D'Ancona, P. and Spagnolo, S., ‘Global solvability for the degenerate Kirchhoff equation with real analytic data’, Invent. Math.108 (1992), 247–262.CrossRefGoogle Scholar
[7]
[7]D'Ancona, P. and Spagnolo, S., ‘On an abstract weakly hyperbolic equation modelling the nonlinear vibration string’, in Developement in partial differential equations and applications to mathematical physics, (Buttazo, G. and Galdi, G. P., Editors) (Plenum Press, New York, 1992), pp. 27–32.CrossRefGoogle Scholar
[8]
[8]Dickey, R.W., ‘Infinite systems of nonlinear oscillation equation with linear damping’, SIAM J. Appl. Math.19 (1970), 208–214.CrossRefGoogle Scholar
[9]
[9]Ebihara, Y., Medeiros, L.A. and Miranda, M.M., ‘Local solutions for a nonlinear degenerate hyperbolic equation’, Nonlinear Anal.10 (1986), 27–40.CrossRefGoogle Scholar
[10]
[10]Kirchhoff, G., Vorlesungen über Mechanik (Teubner, Stuttgart, 1883).Google Scholar
[11]
[11]Medeiros, L.A. and Miranda, M.M., ‘Solutions for the equation of nonlinear vibrations in Sobolev spaces of fractionary order’, Comput. Appl. Math.6 (1987), 257–267.Google Scholar
[12]
[12]Menzala, G.P., ‘On classical solutions of a quasilinear hyperbolic equation’, Nonlinear Anal.3 (1979), 613–627.CrossRefGoogle Scholar
[13]
[13]Mizumachi, T., ‘Decay properties of solutions to degenerate wave equations with dissipative terms’, Adv. Differential Equations4 (1997), 573–592.Google Scholar
[14]
[14]Nakao, M., ‘A difference inequality and its application to nonlinear evolution equations’, J. Math. Soc.Japan30 (1978), 747–762.Google Scholar
[15]
[15]Nishihara, K., ‘On a global solution of some quasilinear hyperbolic equation’, Tokyo J. Math.7 (1984), 437–459.CrossRefGoogle Scholar
[16]
[16]Nishihara, K. and Yamada, Y., ‘On global solutions of some degenerate quasilinear hyperbolic equations with dissipative terms’, Funkcial. Ekvac.33 (1990), 151–159.Google Scholar
[17]
[17]Pozohaev, S.I., ‘On a class of quasilinear hyperbolic equations’, Math. USSR-Sb.25 (1975), 145–158.CrossRefGoogle Scholar
[18]
[18]Rivera, J.E.M., ‘On local strong solutions of a nonlinear PDE’, Appl. Anal.10 (1980), 93–104.CrossRefGoogle Scholar