On the propagation properties of a nonlinear degenerate parabolic equation

Abstract

In this expository paper the equation (1) ut=div(|∇u|m−1 ∇u) is discussed in one space dimension (N=1). L∞-gradient bounds have been established for arbitrary N by the reviewer and R. Rostamian [Math. Ann. 259 (1982), no. 1, 53–70; Proc. Roy. Soc. Edinburgh Sect. A 91 (1981/82), 335–346], and Theorem 1 in the present paper is a special case. The regularity question has been settled for arbitrary N and for general weak solutions by the reviewer and L. C. Evans ["Continuity for the gradient of the solutions of certain degenerate parabolic equations'', J. Math. Pures Appl. (9), to appear]. We note that for N=1, the case considered here, the space derivative of the solution satisfies the porous medium equation, and so results for the solution to that equation translate into results for the derivative of the solutions of (1). Finally we note that, for semigroup solutions f of (1) the generation theorem in the L∞ setting implies immediately the continuity of the solutions. This observation is due to M. Pierre.