Global attractors for quasilinear parabolic–hyperbolic equations governing longitudinal motions of nonlinearly viscoelastic rods

• Attractors for the one-dimensional motion of nonlinearly viscoelastic rods.
• Large longitudinal motions of deformable rods carrying end masses.
• Dependence of dimensions of attractors on mass ratios.
• Quasilinear parabolic–hyperbolic equations.
• Stick–slip frictional forces.

We prove the existence of a global attractor and estimate its dimension for a general family of third-order quasilinear parabolic–hyperbolic equations governing the longitudinal motion of nonlinearly viscoelastic rods carrying an end mass and subject to interesting body forces. The simplest version of the equations has the form wtt=n(wx,wxt)xwtt=n(wx,wxt)x where nn is defined on (0,∞)×R(0,∞)×R and is a strictly increasing function of each of its arguments, with n→−∞n→−∞ as its first argument goes to 0. This limit characterizes a total compression, a source of technical difficulty, which new delicate a priori estimates prevent. We determine how the dimension of the attractor varies with the ratio of the mass of the rod to that of the end mass, giving conditions ensuring that the dimension is small. The estimates of dimension illuminate asymptotic analyses of the governing equation as this mass ratio goes to 0.