Finite-time blowup for the Schrodinger equation with nonlinear source term and for the nonlinear wave equation
报告题目(Title)：Finite-time blowup for the Schrodinger equation with nonlinear source term and for the nonlinear wave equation
报告人(Speaker)：Thierry Cazenave 教授 (法国巴黎六大)
地点(Place)：后主楼1124
时间(Time)：2019年10月24日 15:00-16:00
邀请人(Inviter)：徐桂香
报告摘要
In this joint work with Zheng Han, Yvan Martel and Lifeng Zhao, we consider the nonlinear Schrodinger equation
$\partial _t u = i \Delta u + | u |^\alpha u$ 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 $\partial _{ tt }u = \Delta u + |u|^\alpha u$, which has a comparable structure (finite-time blowup for the associated ODE, and time-reversibility).
主讲人简介
Thierry Cazenave教授是法国巴黎六大教授，与同门Fields奖得主P.L.Lions教授合作利用集中紧方法研究孤子解稳定性、用Strichartz估计研究薛定谔方程局部适定性理论等领域作出杰出工作。其研究专著《Semiclassical Schrodinger Equations》已成为薛定谔方程理论研究经典之作。
