On the growth of the interfaces of a nonlinear degenerate parabolic equation

Abstract

The author deals with the propagation of the support of the initial function in the following problem: ut−(um)xx+cun=0 on R×(0,+∞), u(x,0)=u0(x) on R with n≥m>1, c>0; u0 is bounded and has bounded support, and u0≥0. The author proves the following result: If u is a generalized solution to the above problem and ζ(t)=sup{x∈R: u(t,x)>0} then for n=m, ζ(t)≤ζ(2)+Alnt; and for m<n<m+2,ζ(t)≤ζ(2)+Btβ. A similar result holds for ζ(t)=inf{x∈R: u(t,x)>0}.