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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 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.

We study a continuum model for solid films that arises from the modeling of one-dimensional step flows on a vicinal surface in the attachment-detachment-limited regime. The resulting nonlinear partial differential equation, $u_t = -u^2(u^3+α u)_{hhhh}$, gives the evolution for the surface slope $u$ as a function of the local height $h$ in a monotone step train. Subject to periodic boundary conditions and positive initial conditions, we prove the existence, uniqueness and positivity of global strong solutions to this PDE using two Lyapunov energy functions. The long time behavior of $u$ converging to a constant that only depends on the initial data is also investigated both analytically and numerically.

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