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\(C^{\infty}\)-regularity for the porous medium equation. (English) Zbl 0597.35107

This paper studies the equation \(u_ t=(u^ m)_{xx}\), \(x\in {\mathbb{R}}\), \(t>0\), \(m>1\), for continuous, positive initial data \(u(x,0)=u_ 0(x)\), \(u_ 0\) having compact support. The authors state the known result that \(\sup p u(.,t)=[r(t),s(t)],\) where r’ can have a jump for t equal to \(t_ r:=\sup \{t:\quad r(t)=r(0)\}.\)
Then, working with \(v=u^{m-1}\), they establish the optimal regularity result \(v\in C^{\infty}(\Omega_ r)\), \(r\in C^{\infty}(t_ r,\infty)\), where \(\Omega_ r=\{(x,t):\quad r(t)\leq x<s(t),\quad t>t_ r\}.\)
Reviewer: B.Straugham

MSC:

35Q99 Partial differential equations of mathematical physics and other areas of application
76S05 Flows in porous media; filtration; seepage
35B65 Smoothness and regularity of solutions to PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
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References:

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