Decay Rates Of Plane Viscous Rarefaction Wave For Scalar Conservation Law With Degenerate Viscosity In Several Dimensions

We study the stability problem of the planar waves to the n-dimensional viscousconservation laws with degenerate viscosity in the half spacewith the initial datawhere the boundary condition is prescribed asin which u_- is a constant andAssume that there exists aconstant u₊ such that the initial data satisfyingand the compatibility conditionFurthermore, we assume that the equation (4.2.1) is genuinely nonlinear in the x₁-direction, cf. [20], i.e., there exists a positive constantβ, such thatand that the characteristic speeds f'₁(u_±) satisfywhich implies that u_-< u₊ from (4.2.6) and (4.2.7).Now we can state the results on the asymptotic behavior as follows:Theorem 4.2.2. For any 1≤p <∞, we haveand for any 2≤p <∞where Furthermore, for p=∞, we havefor anyε> 0, where C_εis a positive constant depending onε.We consider the Cauchy problem for the n-dimensional viscous conservation laws with degenerate viscosity in the full space. Now we can state the results on the asymptotic behavior as follows:Theorem 5.2.2. For any 1≤p <∞, we haveand for any 2≤p <∞where Furthermore, for p =∞, we havefor anyε> 0 and for sufficiently large t, where C_εis a positive constant depending onε.The analysis based on the new L^p-energy method.