How long does it take for a gas to fill a porous container?

Abstract:

Let us consider the problem ${u_t}(x,t) = \Delta {u^m}(x,t)$ for $(x,t) \in D \times [0, + \infty ),u(x,0) = {u_0}(x)$ for $x \in D$, and $(\partial {u^m}/\partial n)(x,t) = h(x,t)$ for $(x,t) \in \partial D \times [0, + \infty )$. Here we assume $D \subset {R^N},m > 1,{u_0} \geq 0$, and $h \geq 0$. It is well known that solutions to this problem have the property of finite speed propagation of the perturbations. By this we mean that if z is an interior point of D and exterior to the support of ${u_0}$, then there exists a time $T(z) > 0$ so that $u(z,t) = 0$ for $t < T(z)$ and $u(z,t) > 0$ for $t > T(z)$. In this note we give, in an elementary way, an upper bound for $T(z)$ for the case of bounded convex domains and in the case of a half space.