An Upper Bound for the Waiting Time for Nonlinear Degenerate Parabolic Equations

Abstract

An upper bound is obtained for the time when the support of the solution of some nonlinear, degenerate parabolic equations begins to spread. Solutions to many nonlinear degenerate diffusion equations which are compactly supported initially, remain so at future times (see (4, 7, 8)). We obtain here an upper bound for the time when the support begins to spread, the so-called waiting time. In one space dimension such an upper bound was found by B. F. Knerr (7) who derived a weak ordinary differential equation along the interface (the boundary of the support) and used it to show that the waiting time depends on the size of the initial data near the interface (see also (2, 4, 11, 12)). We replace the equation along the interface with a first order differential in- equality for a local average of the solution near a point on the interface. This allows estimation of the waiting time in higher dimensions, while in one dimension it reproduces previous results (4, 7). In the particular case of the porous medium equation, similar results can also be obtained from a new and sharp Harnack inequality due to D. G. Aronson and L. A. Caffarelli (3). N. D. Alikakos (1) has recently supplied a necessary and sufficient condition for zero waiting time in higher dimensions which relies, in part, upon this inequality. Nevertheless, even for the case of the porous medium equation, our results are elementary and self-contained. Finally, we would like to thank J. L. Vasquez for an informative conversation with the second author and for suggesting that we seek a local version of our preliminary results. 1. Preliminaries. Let uq and g be nonnegative functions. A function u(x,t) is said to be a weak solution of the nonlinear degenerate parabolic equation ut = Ag(u), u(x,0) = uo(x) if, for some T > 0, u satisfies: