QUASI-LINEAR HYPERBOLIC-PARABOLIC EQUATIONS OF ONE-DIMENSIONAL VISCOELASTICITY

We study the global existence of solutions of initial-boundary-value p roblems 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 ana lyses, because they lead to distinctive physical effects. We employ a constitutive equation giving the stress as a general nonlinear Functio n of the strain and the strain rate. All global analyses of this and r elated problems, except that of Dafermos (J. Differential Equations (1 969), 71-86), have employed a stress that is merely affine in the stra in rate. Dafermos's assumptions are far more appropriate for shearing motions than for longitudinal motions. Our constitutive equation satis fies the physically natural requirement that an infinite amount of com pressive 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 ac ute when time-dependent Dirichlet data are prescribed. The further nov el, yet physically reasonable, restrictions we impose on the constitut ive function yield estimates that preclude a total compression anywher e at any finite time. The resulting estimates are crucial for the glob al existence theory we obtain. (C) 1996 Academic Press, Inc.