Limiting Profiles in Contaminant Transport Through Porous Media
C. J. van Duyn, J.Martien de Graaf · SIAM Journal on Mathematical Analysis · 1987
In this paper the following degenerate nonlinear diffusion problem is investigated: \[ \beta (u)_t + u_x = u_{xx} ,\quad t > 0,\quad - \infty 0)$ the equation describes the one-dimensional transport of contaminant in a fluid flow through a homogeneous saturated porous medium. Here the large time behaviour of the solution of the above problem is studied for more general $\beta $. It turns out that, depending upon the shape of $\beta $ (convex or concave) and the values $u_0 ( - \infty )$ and $u_0 ( + \infty )$, the solution converges to a travelling wave g of the form $g(x - at)$ or to a function $\omega ^* $ of the form $\omega ^* ({x / {(t + 1)}})$.