= u_- + \displaystyle{
        {\mathlarger{\int}}^{\frac{\mathlarger{x-\tilde{\lambda} \, t}}
        { \mathlarger{t}^{{\scriptscriptstyle \frac{1}{p+1}}} }}_{-\infty}
                             \Bigl( \left( \, 
                A-B \xi^2 \, 
                \right)\vee 0 \, \Bigr)^{\frac{1}{p-1}}
                \, \mathrm{d}\xi }\end{aligned}$$ which is said to be “contact wave for $p$-Laplacian type viscosity”. Now we are ready to state our main result.

[**Theorem 1.1**]{} (Main Theorem)[**.**]{}

The main theorem is proved by using a technical energy method with the aid of the maximum principle, and the careful estimates of the nonlinear interactions between the nonlinear waves, that is, the rarefaction waves and the contact wave for the $p$-Laplacian type viscosity. This paper is organized as follows. In Section 2, we shall prepare the basic properties of the rarefaction wave and the contact wave for $p$-Laplacian type viscosity. In Section 3, we reduce the problem to an essential case (similarly in [@matsumura-yoshida], [@yoshida]) and reformulate the problem in terms of the deviation from the asymptotic state, that is, the superposition of the nonlinear waves. Following the arguments in [@matsu-nishi2], we show the global existence of the solution to the reformulated problem and the energy estimates which are depending on the time. In order to show the asymptotics, in Section 4 and Section 5, we establish the uniform energy estimates 