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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 $$\Delta$$ $$p\to 0$$.