1. INTRODUCTION Consider a system of viscous conservation laws (1.1) ut + f(u)x = {B{u)ux)x, where u = u(x,t) is the density of physical quantities, f(u) is the flux and B(u) is the viscosity matrix. Many physical models are of this form, e.g. the compressible Navier- Stokes equations, the equations of magnetohydrodynamics and nonlinear viscoelasticity models. We are interested in the explicit, pointwise large time behavior of the solution of the initial value problem of the system. The solution approaches, time-asymptotically, a linear superposition of linear heat kernels, generalized Burgers solutions, and diffusion waves of algebraic types. Several features of the system contribute to the rich qualitative behavior of the solution. The first basic feature is the nonlinearity of the flux function f(u). Physical models are symmetrizable, which implies that the inviscid system (1.2) ut + f(u)x=0 is completely hyperbolic with real characteristics Xi(u) \2{u) • • • Xn(u): f{u)Ti{u) = \i(u)ri(u), li(u)f'(u) = Xi(u)li(u), (1.3) li(u)rj(u) = 6ij, i,j = l , 2 , . . . , n . The first important nonlinearity is the behavior of the characteristic speed Xi(u) in the wave direction ri{u). An z-field is genuinely nonlinear (g.nl.) if Xi(u) is strictly monotone, linearly degenerate (l.dg.) if it is stationary, [La], (g.nl.) V\i(u)'ri(u)^0, (l.dg.) VA i (w).r (u)=0 . Waves pertaining to a (g.nl.) field either compress or expand. Compression waves lead to shock waves. A (l.dg.) field gives rise to linear hyperbolic waves, the contact discontinuities. Received by the editor August 29, 1994 and in revised form October 30, 1995. 1

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