Quasilinear Hyperbolic{parabolic Equations of One{dimensional Viscoelasticity
نویسنده
چکیده
We study the global existence of solutions of initial-boundary-value problems for a quasilinear hyperbolic-parabolic equation describing the longitudinal motion of a one-dimensional viscoelastic rod. We treat a variety of nonhomogeneous boundary conditions, requiring separate analyses, because they lead to distinctive physical eeects. We employ a constitutive equation giving the stress as a general nonlinear function of the strain and the strain rate. All global analyses of this and related problems, except that of Dafermos 13], have employed a stress that is merely aane in the strain rate. Dafermos's assumptions are far more appropriate for shearing motions than for longitudinal motions. Our constitutive equation satisses the physically natural requirement that an innnite amount of compressive stress is needed to produce a total compression at any point of the rod. This requirement is the source of a severe singularity in the governing partial diierential equations, which is particularly acute when time-dependent Dirichlet data are prescribed. The further novel, yet physically reasonable, restrictions we impose on the constitutive function yield estimates that preclude a total compression anywhere at any nite time. The resulting estimates are crucial for the global existence theory we obtain.
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