Regularity of the free boundary for the porous medium equation. (English) Zbl 0621.35101

Consider the Cauchy problem \(u_ t=(u^ m)_{xx}\), \(x\in {\mathbb{R}}\), \(t>0\), \(u(x,0)=u_ 0(x)\), with \(m>1\) and \(u_ 0\geq 0\), supported on a bounded interval. Let \(x=z_{\pm}(t)\) be the boundaries of the support of u(x,t) and set \(v=mu^{m-1}\). The authors prove that \(z_{\pm}(t)\) are \(C^{\infty}\) functions for t in (0,T], with a sufficiently small t, when \(v(x,0)=(1-x^ 2)/a(x)\), a(x) being a positive function in \(C^{J+2}[-1,1],\) \(J=5+2[(2m-1)/(m-1)].\)
The transformation (y,t)\(\to (x(y,t),t)\) is introduced via \[ x_ t(y,t)=-v_ x(x(y,t),t)/(m-1),\quad x(y,0)=y. \] It is observed that \(z_{\pm}(t)=x(\pm 1,t)\) and that the function \(w=(x_ y)^{-m}\) satisfies an initial value problem for a degenerate parabolic equation. The regularity result is then proved by obtaining uniform estimates for Galerkin approximations \(W^{(N)}(y,y)\) to w(y,t).
Reviewer: A.Fasano


35R35 Free boundary problems for PDEs
76S05 Flows in porous media; filtration; seepage
35K65 Degenerate parabolic equations
35D10 Regularity of generalized solutions of PDE (MSC2000)
Full Text: DOI