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**The initial trace of a solution of the porous medium equation.**
*(English)*
Zbl 0556.76084

Let u(x,t) be a continuous weak solution of the porous medium equation \(\partial u/\partial t=\Delta (u^ m)\) in \({\mathbb{R}}^ d\times (0,T]\) for some \(T>0\). We show that corresponding to u there is a unique nonnegative Borel measure \(\rho\) on \({\mathbb{R}}^ d\) which is the initial trace of u. Moreover, we show that the initial trace \(\rho\) must belong to a certain growth class. Roughly speaking, this growth restriction shows that there are no solutions of the porous medium equation whose pressure grows, on average, more rapidly then \(| x|^ 2\) as \(| x| \to \infty\).

### MSC:

76S05 | Flows in porous media; filtration; seepage |

35Q99 | Partial differential equations of mathematical physics and other areas of application |

### Keywords:

continuous weak solution; porous medium equation; unique nonnegative Borel measure; initial trace; growth class
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\textit{D. G. Aronson} and \textit{L. A. Caffarelli}, Trans. Am. Math. Soc. 280, 351--366 (1983; Zbl 0556.76084)

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### References:

[1] | D. G. Aronson, Widder’s inversion theorem and the initial distribution problems, SIAM J. Math. Anal. 12 (1981), no. 4, 639 – 651. · Zbl 0483.35080 |

[2] | Donald G. Aronson and Philippe Bénilan, Régularité des solutions de l’équation des milieux poreux dans \?^{\?}, C. R. Acad. Sci. Paris Sér. A-B 288 (1979), no. 2, A103 – A105 (French, with English summary). · Zbl 0397.35034 |

[3] | Donald Aronson, Michael G. Crandall, and L. A. Peletier, Stabilization of solutions of a degenerate nonlinear diffusion problem, Nonlinear Anal. 6 (1982), no. 10, 1001 – 1022. · Zbl 0518.35050 |

[4] | Philippe Bénilan, Michael G. Crandall, and Michel Pierre, Solutions of the porous medium equation in \?^{\?} under optimal conditions on initial values, Indiana Univ. Math. J. 33 (1984), no. 1, 51 – 87. · Zbl 0552.35045 |

[5] | G. I. Barenblatt, On some unsteady motions of a liquid and gas in a porous medium, Akad. Nauk SSSR. Prikl. Mat. Meh. 16 (1952), 67 – 78 (Russian). · Zbl 0049.41902 |

[6] | Luis A. Caffarelli and Avner Friedman, Regularity of the free boundary of a gas flow in an \?-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), no. 3, 361 – 391. · Zbl 0439.76085 |

[7] | A. S. Kalašnikov, The Cauchy problem in the class of increasing functions for equations of non-stationary filtration type, Vestnik Moskov. Univ. Ser. I Mat. Meh. 1963 (1963), no. 6, 17 – 27 (Russian, with English summary). |

[8] | M. Ughi, Initial values of nonnegative solutions of filtration equation, J. Differential Equations 47 (1983), no. 1, 107 – 132. · Zbl 0476.35049 |

[9] | Björn E. J. Dahlberg and Carlos E. Kenig, Nonnegative solutions of the porous medium equation, Comm. Partial Differential Equations 9 (1984), no. 5, 409 – 437. · Zbl 0547.35057 |

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