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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.

##### MSC:
 35Q72 Other PDE from mechanics (MSC2000) 74K05 Strings 35B40 Asymptotic behavior of solutions to PDEs
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##### References:
 [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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