Regularity of the Interfaces with Sign Changes of Solutions of the One-Dimensional Porous Medium Equation

Abstract

In previous papers we considered the Cauchy problem for the one-dimensional evolution p-Laplacian equation for nonzero, bounded, and nonnegative initial data having compact support, and showed that after a finite time the set of spatial critical points of the nonnegative solution u=u(x, t) in {u>0} consists of one point, the spatial maximum point of u, and the curve of the spatial maximum points is continuous with respect to the time variable. Since the spatial derivative ∂xu satisfies the porous medium equation with sign changes, the curve of the spatial maximum points is regarded as an interface with sign changes of ∂xu. On the other hand, in a paper by M. Bertsch and D. Hilhorst (1991, Appl. Anal.41, 111–130) the interfaces where the solutions change their sign were studied in detail for the initial-boundary value problems of the generalized porous medium equation over two-dimensional cylinders. But the monotonicity of the initial data is assumed there. As is noted in Section 4 of our earlier work (1996, J. Math. Anal. Appl.203, 78–103), the monotonicity of ∂xu(ċ, t) in some neighborhood of the spatial maximum point of u(ċ, t) cannot be assumed, and therefore, if this monotonicity for some large t>0 is proved, then by the method of Bertsch and Hilhorst (cited above) one may get more precise regularity properties of the curve of the spatial maximum points. The purpose of the present paper is twofold. One is to remove some monotonicity assumption for initial data in Bertsch and Hilhorst's theorem concerning the regularity of the interfaces with sign changes of solutions of the one-dimensional generalized porous medium equation. By comparing the solution with appropriate symmetric nonnegative solutions we shall get the monotonicity of the solution near the interface after a finite time. The other is as a by-product of the method to get C1 regularity of the curves of the spatial maximum points of nonnegative solutions of the Cauchy problem for the evolution p-Laplacian equation for sufficiently large t.