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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 ${u}_{t}-{\left({u}^{m}\right)}_{xx}=0$, $x\in R$, $t\ge 0$. By the transformation $v=m{u}^{m-1}/\left(m-1\right)$ this equation goes over to

$\left(1\right)\phantom{\rule{1.em}{0ex}}{v}_{t}-\left(m-1\right)v{v}_{xx}-{v}_{x}^{2}=0$

and the authors show that for $m↓1$ solution v converges to the solution w of (2) ${w}_{t}-{\left({w}_{x}\right)}^{2}=0$. Both equations (1) and (2) have the finite speed of propagation of disturbances.

Reviewer: O.Vejvoda
##### MSC:
 35K55 Nonlinear parabolic equations 35F20 General theory of first order nonlinear PDE 76S05 Flows in porous media; filtration; seepage 35B40 Asymptotic behavior of solutions of PDE