Cauchy Problem For The Nonhomogeneous Hyperbolic Conservation Laws With The Degenerate Viscous Term

In this paper, we consider one-dimensional or multi-dimensional Cauchy problem for the nonhomogeneous hyperbolic conservation laws with the degenerate viscous term.The whole paper is departed to two parts:In the the first part, we consider one-dimensional Cauchy problem for the non-homogeneous hyperbolic conservation laws with the degenerate viscous term:where here f(u), g(u) is a one order continuous and differentiable function defined on R, a > 0, 0 <α< 1 are both constants, under these conditions, we will study the Cauchy problem from the following aspects. First, we give the similar solution of the Cauchy problem for the linear and homogenous equation corresponding to (I), then make the appropriate sequence of the representation formula to abstain the local existence of solutions. Last, we get L~∞estimate of solution by the maximum principle and make use of the extension theorem to obtain the global existence.In the second parts, we consider multi-dimensional Cauchy problem for the nonhomogeneous hyperbolic conservation laws with the degenerate viscous term:where f_i(u), F(u), i = 1,2,3…are one-order smooth functions defined on R~n, a > 0, 0 <α< 1 are both constants. We can obtain the global existence similarly.