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Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity. (English) Zbl 0844.35021

The aim of this paper is to give sufficient conditions for a nonlinear stress function \(n= n(s, w_s, w_{st})\) guaranteeing global-in-time existence of solutions of initial-value problem for the quasilinear hyperbolic-parabolic equation \(\varrho w_{tt}= n(s, w_s, w_{st})_s+ f(s, t)\) describing longitudinal motions of a viscoelastic rod. Here \(w= w(s, t)\), \((s, t)\in (0, 1)\times \mathbb{R}^+\) denotes the position of a material point and therefore \(w_s(s, t)\) is the stretch at \((s, t)\). The function \(n= n(s, y, 0)\) is allowed to diverge to \(-\infty\) as \(y\to 0\). This causes the above equation to be singular. The proof of global existence of solutions is based on delicate estimates showing that total compression (i.e. the stretch \(w_s(s, t)\) vanishes) cannot occur in finite time.

MSC:

35G25 Initial value problems for nonlinear higher-order PDEs
74Hxx Dynamical problems in solid mechanics
74K10 Rods (beams, columns, shafts, arches, rings, etc.)
35K20 Initial-boundary value problems for second-order parabolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
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