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Quasilinear equation in moving domain. (English) Zbl 1070.35109
Summary: We are concerned with the existence and uniqueness of global weak solutions of a mixed problem associated with the one-dimensional damped elastic stretched string equation \[ u_{tt}(x,t)- \biggl( p(t)+q(t)\int_{\Omega_t} |u_x(x,t)|_{\mathbb R}^2\, dx\biggr) u_{xx}(x,t)- \delta u_{xxt}(x,t)= 0 \quad\text{in }Q_t, \] when the supports of the ends have small displacements. In addition, we show that the energy decays exponentially. In previous investigations about the string equation in a moving domain, local or global solutions for an increasing in time domain has been shown. Here, thanks to the internal strong damping we eliminate this hypothesis.

35Q72 Other PDE from mechanics (MSC2000)
74K05 Strings
35B40 Asymptotic behavior of solutions to PDEs
Full Text: DOI
[1] Arosio, A.; Spagnolo, S., Global solutions to the Cauchy problem for a nonlinear hyperbolic equation, () · Zbl 0598.35062
[2] Brito, E.H., The damped elastic stretched string equation generalized: existence, uniqueness, regularity and stability, Appl. anal., 13, 219-233, (1982) · Zbl 0458.35065
[3] Clark, H.R., Global classical solutions to the Cauchy problem for a nonlinear wave equation, Int. J. math. math. sci., 21, 3, 533-548, (1998) · Zbl 0908.35080
[4] Clark, H.R., Asymptotic and smoothness properties of a nonlinear equation with damping, Commun. appl. anal., 4, 3, 321-337, (2000) · Zbl 1084.34537
[5] Dickey, R.W., The initial value problem for a nonlinear semi-infinite string, Proc. roy. soc. Edinburgh, 82A, 19-26, (1978) · Zbl 0394.45007
[6] A. Haraux, E. Zuazua, Decay estimates for some semilinear damped hyperbolic problems, Arch. Rat. Mech. Anal. (1988) 191-206. · Zbl 0654.35070
[7] Kirchhoff, G., Vorlesungen über mechanik, (1883), Tauber Leipzig · JFM 08.0542.01
[8] Límaco, J.; Medeiros, L.A., Kirchhoff – carrier elastic strings in non-cylindrical domains, Port. math., 56, 4, 465-500, (1999) · Zbl 0943.35001
[9] Lions, J.L., Une remarque sur LES problèmes d’evolution non linéaires dans des domaines non cylindriques, Rev. romaine math. pure appl., 9, 11-18, (1964) · Zbl 0178.12302
[10] Lions, J.L., Quelques méthodes de résolution des problèmes aux limites non linéaires, (1969), Dunod Paris · Zbl 0189.40603
[11] Matos, M.P., Mathematical analysis of the nonlinear model for the vibrations of a string, Nonlinear anal., 17, 12, 1125-1137, (1991) · Zbl 0763.35061
[12] Matos, M.P.; Pereira, D.C., On a hyperbolic equation with strong damping, Funkcialaj ekvacioj, 34, 303-311, (1991) · Zbl 0746.34039
[13] Medeiros, L.A., On a new class of nonlinear wave equations, J. math. anal. appl., 69, 1, 252-262, (1979) · Zbl 0407.35051
[14] Medeiros, L.A.; Límaco, J.; Menezes, S.B., Vibrations of elastic strings: mathematical aspects, part 1, J. comp. anal. appl., 4, 2, 91-127, (2002) · Zbl 1118.35335
[15] Medeiros, L.A.; Límaco, J.; Menezes, S.B., Vibrations of elastic strings: mathematical aspects, part two, J. comp. anal. appl., 4, 3, 211-263, (2002) · Zbl 1118.35336
[16] L.A. Medeiros, M. Milla Miranda, On a nonlinear wave equation with damping, Rev. Mat. Univ. Complutense Madrid, vol. 3, No. 2 y 3, 1990, pp. 213-231. · Zbl 0721.35044
[17] Nishihara, K., Degenerate quasilinear hyperbolic equation with strong damping, Funcialaj ekvacioj, 27, 125-145, (1984) · Zbl 0555.35094
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