关键词: 双曲守恒律, 退化粘性, 极值原理, L^∞估计, 整体存在性

Abstract: In this paper, the authors consider Cauchy problem for the nonhomogeneous hyperbolic conservation
laws with the degenerate viscous term
$$
\left\{\begin{array}{l}
u_t+f(u)_x=a^2t^\alpha u_{xx}+g(u),\ \ \ x\in{\bf R},\ \ \ t>0,\\
u(x,0)=u_0(x) \in L^\infty({\bf R}).
\end{array}\right.
\eqno{({\rm I})}
$$
where here $f(u),g(u)$ is a one order continuous and differentiable function defined on
${\bf R}, a>0, 0<\alpha <1$ are both constants. Under these conditions, the authors obtain the
local existence of solutions of the Cauchy problem (I). Then, the authors get $L^\infty$ estimate of solution
by the maximum principle and make use of the extension theorem to obtain the global existence.

Key words: Hyperbolic conservation laws, Degenerate viscosity, Maximum principle, L^∞ estimate, Global existence