报告题目：Finite-time blowup for a Schroedinger equation with a nonlinear source term

报告人:   Thierry  Cazenave 教授（法国索邦大学）

时间：2019年10月16日下午13:30-15:00

地点：浙江大学玉泉校区欧阳楼314

摘要: In this joint work with Zheng Han, Yvan Martel and Lifeng Zhao, we consider the nonlinear Schr/"odinger equation $u_t = i /Delta u + | u |^/alpha u$ on ${/mathbb R}^N $, for $H^1$-subcritical or critical nonlinearities: $/alpha>0$ and $(N-2) /alpha /le 4$. This equation combines two important properties: the associated ODE $u'= | u |^/alpha u$ produces finite-time blowup; and the equation can be solved backwards in time. Using these properties we prove that, given any compact set $ E /subset {/mathbb R}^N $, there exist finite-energy solutions which are defined on some time interval $(-T, 0)$ and blow up at $t=0$ exactly on $ E$. The construction is based on an appropriate ansatz. The initial ansatz (which is sufficient when $/alpha >1$) is simply $U_0(t,x) = /kappa  (t + A(x) )^{ -/frac {1} {/alpha } }$, where $A/ge 0$ vanishes exactly on $ E$, which is a solution of the ODE $u'= | u |^/alpha u$. If $/alpha /le 1$, we need to refine this ansatz, and we proceed inductively, using only ODE techniques. We complete the proof by energy estimates and a compactness argument. We prove similar results for the nonlinear wave equation, which has a comparable structure (finite-time blowup for the associated ODE, and time-reversibility).

报告人简介： Thierry  Cazenave 现任法国索邦大学(巴黎第六大学)教授:分别于1975年和1984年获得该校硕士学位和博士学位，并留校任教至今。 Cazenave Thierry Thierry  Cazenave 教授是当今世界上非线性发展方程领域的领军人物，其研究工作主要涉及(色散或非色散)双曲型和抛物型问题。与 P.L. Lions，J. Shatah，F. Merle，H. Blezis等著名数学家均有合作研究。在 CMP、TAMS、 JFA等世界著名杂志上发表论文70多篇，并被广泛引用。

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联系人：张挺 (zhangting79@zju.edu.cn)

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