Existence of a global solution for a viscoelastic system

In this paper, a unique classical solution (v, u) of the Cauchy problem (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 obtained 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-Verlag, New York/Berlin, 1982) for The viscoelastic system. A parallel result (Theorem 8) for a model (42) of compressible adiabatic flow through porous media with a physical viscosity is also obtained. (C) 1998 Academic Press.