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Zbl 0635.35047
Aronson, D.G.; Vazquez, J.L.
The porous medium equation as a finite-speed approximation to a Hamilton- Jacobi equation. (English)
[J] Ann. Inst. Henri Poincaré, Anal. Non Linéaire 4, 203-230 (1987). ISSN 0294-1449

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.
[O.Vejvoda]

MSC 2000:: *35K55 Nonlinear parabolic equations
35F20 General theory of first order nonlinear PDE
76S05 Flows in porous media
35B40 Asymptotic behavior of solutions of PDE

Keywords: porous medium flow; Hamilton-Jacobi equation; viscosity solutions interfaces; finite speed; propagation of disturbances

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