EXISTENCE OF A GLOBAL SOLUTION FOR A VISCOELASTIC SYSTEM

In this paper, a unique classical solution (v, u) of the Cauchy proble m (1), (2) is obtained if \v(0)(x)\(1+alpha) less than or equal to M, \u(0)(x)\(2+alpha) less than or equal to M. The a priori estimates \v( x, t)\(1+alpha), \u(x, t)\(2+alpha) less than or equal to M(T) are obt ained by using the maximum principle without the restriction lim(\x\-- >+/-infinity) (v(0)(x), u(0)(x)) = (v(+/-), u(+/-)), where v(+/-), u(/-) are constants. This resolves a problem proposed by J. A. Smeller ( ''Shock Waves and Reaction-Diffusion Equations,'' p. 444, Springer-Ver lag, New York/Berlin, 1982) for The viscoelastic system. A parallel re sult (Theorem 8) for a model (42) of compressible adiabatic flow throu gh porous media with a physical viscosity is also obtained. (C) 1998 A cademic Press.