Quasilinear Hyperbolic–Parabolic Equations of One-Dimensional Viscoelasticity

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 effects. 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 (J. Differential Equations(1969), 71–86), have employed a stress that is merely affine in the strain rate. Dafermosʹs assumptions are far more appropriate for shearing motions than for longitudinal motions. Our constitutive equation satisfies the physically natural requirement that an infinite 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 differential 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 finite time. The resulting estimates are crucial for the global existence theory we obtain.