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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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