Solutions with compact support of the porous medium equation in arbitrary dimensions

Abstract:

Compactness of the support is discussed of a solution $u$ to the Cauchy problem for the porous medium equation ${u_t} = \Delta \phi (u),t > 0$, in ${R^N}$ of arbitrary dimension $N \geq 1$, where $\phi$ is a nondecreasing function on ${R^1}$. It is shown that if $u(0,x) = 0$ for $\left | x \right | \geq R,R > 0$, then for all $t \geq 0$ \[ u(t,x) = 0\quad {\text {a}}{\text {.e}}{\text {.}}\left | x \right | \geq R + C{t^{1/2}}\] with a constant $C$ depending on $\phi$ and $u(0, \cdot )$. The result is well known when $N = 1$, but the study for $N > 1$ has somehow been neglected.