Hölder continuity of interfaces for the porous medium equation with absorption.

*(English)*Zbl 0818.35053From the introduction: This paper is devoted to the porous medium equation with absorption
\[
u_ t = \Delta u^ m - u^ p \quad \text{in } Q
\]
with \(m > 1\) and \(p \geq 1\), where \(Q = \mathbb{R}^ N \times (0, \infty)\). In a recent paper, the author proved some generalized Harnack inequality for bounded weak solutions. On the basis of such kind of the Harnack inequality and its variant, we can prove the Hölder continuity of the interfaces for the solutions and this is the purpose of the present paper.

##### MSC:

35K65 | Degenerate parabolic equations |

35D10 | Regularity of generalized solutions of PDE (MSC2000) |

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

##### Keywords:

porous medium equation with absorption; generalized Harnack inequality; Hölder continuity of the interfaces
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\textit{H. Yuan}, Commun. Partial Differ. Equations 18, No. 5--6, 965--976 (1993; Zbl 0818.35053)

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##### References:

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[2] | Yazhe, C. ”Uniqueness of the generalized solution of quasi-linear degenerate parabolic equation Proceedings of the 1982 Changchun Symposium on differential geometry and differential equations”. 317–332. |

[3] | Hongjun, Y. ”Harnack inequality for bounded weak solutions of the porous medium equation with absorption, to appear”. · Zbl 0879.35086 |

[4] | Hongjun, Y. ”Regularity of free boundaries for general filtration equations, to appear.”. |

[5] | Giaquinta, M. 1983. ”Multiple integrals in the calculus of variations and nonlinear elliptic systems”. Princeton Univ. Press. · Zbl 0516.49003 |

[6] | Ladyzhenskaya, O.A., Ural’tzeva, N. N. and Providence, R. I. ”Linear and quasilinear equations of parabolic type Trans. Math. Monographs”. Edited by: Solonnikov, N.A. |

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