A nonlinear integral equation occurring in a singular free boundary problem

Abstract

Abstract. We study the Cauchy problem j u, = <j>(ux)x, (x, ?) e R x R + , \"(,0)=/ with the piecewise linear constitutive function <J>(|) = £+ = max(0, |) and with smooth initial data/which satisfy xf'(x) > 0, x e R, and/"(0) > 0. We prove that free boundary s, given by ux(s(t)+, t) = 0, is of the formwhere the constant k = 0.9034... is the (numerical) solution of a particular nonlinear equation. Moreover, we show that for any a e (0,1/2), /W'))=o('°"'). '-o+.The proof involves the analysis of a nonlinear singular integral equation.

A Class of One-Dimensional Degenerate Parabolic Equations.

“…In the special case of (P), (g) with the function g satisfying the additional assump tion: g'(0 + ) 9 g'(0~) =t = 0, we have shown by solving a singular integral equation [HN2,3] that the problem (P) has a unique solution (v 9 s). Moreover, The purpose of this note is to discuss the regularity and qualitative behavior for small t of the free boundary of (P) for more general initial data g. Of particular interest is the model datum…”

A class of one-dimensional degenerate parabolic equations

“…In the special case of (P), (g) with the function g satisfying the additional assump tion: g'(0 + ) 9 g'(0~) =t = 0, we have shown by solving a singular integral equation [HN2,3] that the problem (P) has a unique solution (v 9 s). Moreover,…”

“…where p ±9 y > 0 are given constants, for which the integral equation method of [HN2,3] breaks down if y 4= 1, i.e. if the datum g is not piecewise linear.…”