A coordinate transformation for the porous media equation that renders the free-boundary stationary. (English) Zbl 0574.76093

The authors investigate the degenerate diffusion problem (1) \(u_ t=(u^ 2)_{xx},\quad u(0,x)=u_ 0(x)\) with \(u_ 0(x)>0\) for \(x\in (- a,a)\) and \(u_ 0(x)=0\) otherwise. They introduce Lagrangian coordinates X and U where \(x=X(t,p)\) such that \(X(0,p)=0,\quad U=X_ p^{-1}u_ 0\), and (2) \(X^ 3_ pX_ t=2u_ 0X_{pp}-2u_ 0'X_ p\). The purpose of the paper is an analysis of the free boundary \(X(t,\pm a)\). The authors show in particular: (i) there exists at most one regular solution on [0,T) for any \(T>0\), (ii) if \(u''_ 0<0\) on [-a,a], then a solution X is regular on [0,T) for any \(T>0\), (iii) if \(u_ 0'(b)=0\) and \(u_ 0''(b)>0\) for \(b=-a\) or \(b=a\), then X(t,b)\(\equiv b\) for \(t\in [0,T)\) and \(X_ p(t,b)\to 0\) as \(t\to T=(6u_ 0''(b))^{-1}\), (iv) if \(u''_ 0<0\) on [-a,a], the integral of \(X^ 2_ p\) over [-a,a] is bounded by a linear function of t. The authors additionally adopt an explicit-implicit difference scheme for the initial value problem with the PDE (2). They present numerical results for a problem whose solution of (1) is known explicitly and they verify numerically the property \(O(\Delta p^ 2)\) as \(\Delta\) \(p\to 0\).
Reviewer: E.Adams


76R99 Diffusion and convection
76S05 Flows in porous media; filtration; seepage
35Q99 Partial differential equations of mathematical physics and other areas of application
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