Seminar on Differential Equations

主講者: | 史習偉博士 (University of Minnesota) |

講題: | Some results on scattering for log-subcritical and log-supercritical nonlinear wave equations |

時間: | 2012-07-19 (Thu.) 16:00 - 17:00 |

地點: | 數學所 722 研討室 (台大院區) |

Abstract: | We consider two problems in the asymptotic behavior of semilinear second order wave equations. First, we consider the $\dot{H}^1_x\times L^2_x$ scattering theory for the energy log-subcritical wave equation $$\square u={\left|u\right|}^{4}ug\left(\left|u\right|\right)$$ in $\mathbb{R}^{1+3}$, where $g$ has logarithmic growth at $0$. We discuss the solution with general (resp. spherically symmetric) initial data in the logarithmically weighted (resp. lower regularity) Sobolev space. The second problem studied here involves the energy log-supercritical wave equation $$\square u={\left|u\right|}^{4}ulo{g}^{\alpha}\left(2+{\left|u\right|}^{2}\right),for0<\alpha ?\frac{4}{3}$$ in $\mathbb{R}^{1+3}$. We prove the same results of global existence and $(\dot{H}^1_x\cap \dot{H}^2_x)\times H^1_x$ scattering for the equation with a slightly higher power of the logarithm factor in the nonlinearity than that allowed by the previous work of Tao. |

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