The interfaces of one-dimensional flows in porous media

Abstract:

The solutions of the equation ${u_t} = {({u^m})_{x x}}$ for $x \in {\mathbf {R}},0 < t < T,m > 1$, where $u(x,0)$ is a nonnegative Borel measure that vanishes for $x > 0$ (and satisfies a growth condition at $- \infty$), exhibit a finite, monotone, continuous interface $x = \zeta (t)$ that bounds to the right the region where $u > 0$. We perform a detailed study of $\zeta$: initial behaviour, waiting time, behaviour as $t \to \infty$. For certain initial data the solutions blow up in a finite time ${T^{\ast }}$: we calculate ${T^{\ast }}$ in terms of $u(x,0)$ and describe the behaviour of $\zeta$ as $t \uparrow {T^{\ast }}$.