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

$n≥q 1$ |

$M_c=\frac{2\sqrt{6}π}{3}$ |

$n=1$ |

$M_c$ |

$n≥q 4$ |

$n=1$ |

$M_c$ |

$n=1$ |

$M_c$ |

$ h(·, t_k)\rightharpoonup 0$ |

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

${t_k} \to \infty $ |

This paper investigates the existence of a uniform in time $L^{∞}$ bounded weak entropy solution for the quasilinear parabolic-parabolic Keller-Segel model with the supercritical diffusion exponent $0<m<2-\frac{2}{d}$ in the multi-dimensional space ${\mathbb{R}}^d$ under the condition that the $L^{\frac{d(2-m)}{2}}$ norm of initial data is smaller than a universal constant. Moreover, the weak entropy solution $u(x,t)$ satisfies mass conservation when $m>1-\frac{2}{d}$. We also prove the local existence of weak entropy solutions and a blow-up criterion for general $L^1\cap L^{∞}$ initial data.

In this paper, we study the modified Camassa-Holm (mCH) equation in Lagrangian coordinates. For some initial data $m_0$, we show that classical solutions to this equation blow up in finite time $T_{max}$. Before $T_{max}$, existence and uniqueness of classical solutions are established. Lifespan for classical solutions is obtained: $T_{max}≥ \frac{1}{||m_0||_{L^∞}||m_0||_{L^1}}.$ And there is a unique solution $X(ξ, t)$ to the Lagrange dynamics which is a strictly monotonic function of $ξ$ for any $t∈[0, T_{max})$: $X_ξ(·, t)>0$. As $t$ approaching $T_{max}$, we prove that the classical solution $m(·, t)$ in Eulerian coordinates has a unique limit $m(·, T_{max})$ in Radon measure space and there is a point $ξ_0$ such that $X_ξ(ξ_0, T_{max}) = 0$ which means $T_{max}$ is an onset time of collisions of characteristics. We also show that in some cases peakons are formed at $T_{max}$. After $T_{max}$, we regularize the Lagrange dynamics to prove global existence of weak solutions $m$ in Radon measure space.

In this paper, we study 1D autonomous fractional ODEs $D_c^{γ}u=f(u), 0< γ <1$, where $u: [0,∞) \to \mathbb{R}$ is the unknown function and $D_c^{γ}$ is the generalized Caputo derivative introduced by Li and Liu (arXiv:1612.05103). Based on the existence and uniqueness theorem and regularity results in previous work, we show the monotonicity of solutions to the autonomous fractional ODEs and several versions of comparison principles. We also perform a detailed discussion of the asymptotic behavior for $f(u)=Au^p$. In particular, based on an Osgood type blow-up criteria, we find relatively sharp bounds of the blow-up time in the case $A>0, p>1$. These bounds indicate that as the memory effect becomes stronger ($γ \to 0$), if the initial value is big, the blow-up time tends to zero while if the initial value is small, the blow-up time tends to infinity. In the case $A<0, p>1$, we show that the solution decays to zero more slowly compared with the usual derivative. Lastly, we show several comparison principles and Grönwall inequalities for discretized equations, and perform some numerical simulations to confirm our analysis.

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