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DCDS-B

In this paper, we study existence of global entropy weak solutions to a critical-case unstable thin film equation in one-dimensional case

$h_t+\partial_x (h^n\,\partial_{xxx} h)+\partial_x (h^{n+2}\partial_{x} h)=0,$ |

where

. There exists a critical mass

found by Witelski et al.(2004 Euro. J. of Appl. Math. 15,223-256) for

. We obtain global existence of a non-negative entropy weak solution if initial mass is less than

. For

, entropy weak solutions are positive and unique. For

, a finite time blow-up occurs for solutions with initial mass larger than

. For the Cauchy problem with

and initial mass less than

, we show that at least one of the following long-time behavior holds:the second moment goes to infinity as the time goes to infinity or

in

for some subsequence

.

$n≥q 1$ |

$M_c=\frac{2\sqrt{6}π}{3}$ |

$n=1$ |

$M_c$ |

$n≥q 4$ |

$n=1$ |

$M_c$ |

$n=1$ |

$M_c$ |

$ h(·, t_k)\rightharpoonup 0$ |

$L^1(\mathbb{R})$ |

${t_k} \to \infty $ |

DCDS

This note is devoted to the discussion on the existence and blow up of the solutions to the parabolic elliptic type Patlak-Keller-Segel system on the whole space case. The problem in two dimension is closely related to the Logarithmic Hardy-Littlewood-Sobolev inequality, which directly introduced the critical mass $8\pi$. While in the higher dimension case, it is related to the Hardy-Littlewood-Sobolev inequality. Therefore, a porous media type nonlinear diffusion has been introduced in order to balance the aggregation. We will review the critical exponents which were introduced in the literature, namely, the exponent $m=2-2/n$ which comes from the scaling invariance of the mass, and the exponent $m=2n/(n+2)$ which comes from the conformal invariance of the entropy. Finally a new result on the model with a general potential, inspired from the Hardy-Littlewood-Sobolev inequality, will be given.

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