Similarity solutions of the porous medium equation with sign changes.

*(English)*Zbl 0777.35034We study similarity solutions with sign changes of the porous medium equation in one space dimension,
\[
u_ t=(| u|^{m-1} u_ x)_ x,\quad x\in\mathbb{R},\;t>0,\tag{1}
\]
with \(m>1\). Here \(u\) is a function of \(x\) and \(t\). For nonnegative \(u\) eq. (1) arises in the theory of gas flow through a one-dimensional porous medium, and an extensive theory has been developed over the last decades.

For the development of the theory of (1) similarity solutions play an important role. Essentially these are solutions whose profiles remain the same as \(t\) varies. The most familiar of them are the Barenblatt-Pattle solutions and the so-called dipole solutions.

Our result concerns semilarity solutions of the form \(u(x,t)=t^ \alpha U(\eta)\), where \(\eta=xt^{-\beta}\), and \(\alpha\) and \(\beta\) are fixed reals. It is well-known that they have to be related by \(2\beta=(m- 1)\alpha+1\), and that \(U(\eta)\) has to satisfy the second order ordinary differential equation \[ (| U|^{m-1} U)''+m\beta\eta U'=m\alpha U. \] {}.

For the development of the theory of (1) similarity solutions play an important role. Essentially these are solutions whose profiles remain the same as \(t\) varies. The most familiar of them are the Barenblatt-Pattle solutions and the so-called dipole solutions.

Our result concerns semilarity solutions of the form \(u(x,t)=t^ \alpha U(\eta)\), where \(\eta=xt^{-\beta}\), and \(\alpha\) and \(\beta\) are fixed reals. It is well-known that they have to be related by \(2\beta=(m- 1)\alpha+1\), and that \(U(\eta)\) has to satisfy the second order ordinary differential equation \[ (| U|^{m-1} U)''+m\beta\eta U'=m\alpha U. \] {}.

##### MSC:

35K65 | Degenerate parabolic equations |

35Q35 | PDEs in connection with fluid mechanics |

35K57 | Reaction-diffusion equations |

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

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\textit{J. Hulshof}, J. Math. Anal. Appl. 157, No. 1, 75--111 (1991; Zbl 0777.35034)

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##### References:

[1] | Angenent, S.B, Analiticity of the interfaces of the porous media equation after the waiting time, (), 329-336, No. 2 · Zbl 0653.35040 |

[2] | Arnold, V.I, Geometrical methods in the theory of ordinary differential equations, (1983), Springer-Verlag New York/Berlin · Zbl 0507.34003 |

[3] | Aronson, D.G, The porous medium equation, () · Zbl 0626.76097 |

[4] | {\scM. Bertsch and D. Hilhorst}, The interface between regions where u < 0 and u > 0 in the porous medium equation, submitted for publication. · Zbl 0675.76094 |

[5] | van Duijn, C.J; Gomes, S.M; Hongfei, Zhang, On a class of similarity solutions of the equation \(ut = (¦u¦\^{}\{m − 1\} ux)x with m > − 1\), IMA J. appl. math., 41, 147-163, (1988), [Originally Chap. 4 of the third author’s thesis written at the Delft University of Technology, 1988] · Zbl 0701.35090 |

[6] | van Duijn, C.J; Peletier, L.A, On a class of similarity solutions of the nonlinear diffusion equation, J. nonlin. anal., 1, 223-233, (1977) · Zbl 0394.34016 |

[7] | Gilding, B.H; Peletier, L.A, On a class of similarity solutions of the porous media equation, J. math. anal. appl., 55, 351-364, (1976) · Zbl 0356.35049 |

[8] | Gilding, B.H; Peletier, L.A, On a class of similarity solutions of the porous media equation, II, J. math. anal. appl., 57, 522-538, (1977) · Zbl 0365.35029 |

[9] | Gilding, B.H, On a class of similarity solutions of the porous media equation, III, J. math. anal. appl., 77, 381-402, (1980) · Zbl 0454.35053 |

[10] | Hulshof, J, Similarity solutions of the porous medium equation with sign changes, research announcement, Appl. math. newsl., 2, No. 3, 229-232, (1989) · Zbl 0709.35056 |

[11] | Jones, C.W, On reducible nonlinear differential equations occurring in mechanics, (), 327-343 · Zbl 0052.31303 |

[12] | de Josseling de Jong, G; van Duijn, C.J, Transverse dispersion from an originally sharp fresh-salt interface caused by shear flow, J. hydrology, 84, 55-79, (1986) |

[13] | {\scS. Kamin and J. L. Vazquez}, Asymptotic behaviour of solutions of the porous medium equation with changing sign, to appear. · Zbl 0755.35011 |

[14] | {\scS. Kamin and J. L. Vazquez}, private communication. |

[15] | Peletier, L.A, The porous media, equation, () · Zbl 0497.76083 |

[16] | {\scJ. L. Vazquez}, New selfsimilar solutions of the porous medium equation and the theory of solutions with changing sign, J. Nonlin. Anal., in press. · Zbl 0734.35042 |

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