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The porous medium equation as a finite-speed approximation to a Hamilton- Jacobi equation. (English) Zbl 0635.35047
The authors study the initial problem for $u\sb t-(u\sp m)\sb{xx}=0$, $x\in R$, $t\ge 0$. By the transformation $v=mu\sp{m-1}/(m-1)$ this equation goes over to $$ (1)\quad v\sb t-(m-1)vv\sb{xx}-v\sp 2\sb x=0 $$ and the authors show that for $m\downarrow 1$ solution v converges to the solution w of (2) $w\sb t-(w\sb x)\sp 2=0$. Both equations (1) and (2) have the finite speed of propagation of disturbances.
Reviewer: O.Vejvoda

35K55Nonlinear parabolic equations
35F20General theory of first order nonlinear PDE
76S05Flows in porous media; filtration; seepage
35B40Asymptotic behavior of solutions of PDE
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