Asymptotics toward the diffusion wave for a one-dimensional compressible flow through porous media

Consider the Cauchy problem for a one-dimensional compressible flow through porous media, v_t - u_x = 0, x ∈ R, t > 0, u_t + p(v)x = -αu, (v, u)|t=0 = (v₀, u₀) (x). Hsiao and Liu showed that the solution (v, u) behaves as the diffusion wave (v̄, ū), i.e. the solution of the porous-media equation due to the Daroy law. The optimal convergence rates have been obtained by Nishihara and co-workers. When v₀(x) has the same constant state at x = ±∞, the convergence rate ∥(v - v̄)(·, t)∥_L∞ = O(t_-1 obtained is 'optimal', since ∥v̄(·, t)∥∞ = O(t^-1/2). However, this 'optimal' convergence rate is less sufficient to determine the location of the diffusion wave. Our aim in this paper is to obtain the 'truly optimal' convergence rate by choosing suitably located diffusion waves.