## 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
Full Text: