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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

##### MSC:
 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)
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