A B S T R A C T We are interested in the time-asymptotic behavior of solutions to viscous conservation laws. Through the pointwise estimates for the Green's function of the linearized system and the analysis of coupling of nonlinear diffusion waves, we obtain explicit expressions of the time-asymptotic behavior of the solutions. This yields optimal estimates in the integral norms. For most physical models, the viscosity matrix is not positive definite and the system is hyperbolic-parabolic, and not uniformly parabolic. This implies that the Green's function may contain Dirac 6-functions. When the corresponding inviscid system is non-strict ly hyperbolic, the time-asymptotic state contains generalized Burgers solutions. These are illustrated by applying our general theory to the compressible Navier- Stokes equations and the equations of magnetohydrodynamics. 1991 Mathematics Subject Classification: Primary 35K55, 76N10 Secondary 35B40, 35A08, 35L65, 76W05. Key words and phrases: quasilinear hyperbolic-parabolic systems, large time behavior, Green's function, nonlinear and linear diffusion waves, conservation laws, compressible Navier-Stokes equations, magneto- hydrodynamics. The research of the first author was partially supported by Army Research Grant DAAH04-94-G-0045 and NSF Grant DMS-9216275-001. The research of the second author was supported by the Office of Naval Research under contract # N00014-92-J-1481. Additional support was also provided by the Army Research Office under contract # DAAH04-93-G-0125 and by the National Science Foundation under grant # DMS-9307928. This monograph was prepared in A\^S-T^i.
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