Speaker: Prof. Daoyin He (Southeast University)
Time: 9:00-10:00 December 22, 2023 (Friday)
Place: MCM110
Title: Long Time Evolution of Semilinear Tricomi Equations
Abstract: We consider the global Cauchy problem for the semilinear generalized Tricomi equation \partial_t^2u-t^m∆u=up with suitable initial data \left(u\left(0,\bullet\right), \partial_tu\left(0,\bullet\right)\right)=\left(u_0,u_1\right), where t\geq0, x\in\mathbb{R}^n\left(n\geq1\right), m\in\mathbb{N}, p>1. We determine a critical exponent p_{crit}\left(m,n\right)>1 such that the solution u, in general, blows up in finite time when 1<p\le p_{crit}\left(m,n\right), while the global existence result with small data is established for p>p_{crit}\left(m,n\right). This is a joint work with Prof. Yin and Prof. Witt.
Speaker: Prof. Xu Yuan (Chinese University of Hong Kong)
Time: 10:00-11:00 December 22, 2023 (Friday)
Place: MCM110
Title: Quantitative observability for one-dimensional Schrödinger equations with potentials
Abstract: In this talk, we will discuss the observability problem for Schrödinger equations. In the existing literature, the observability of Schrödinger equation on compact manifolds and bounded domains has been extensively studied. However, the observability problem on an unbounded set (by measurable control regions) is much less studied in the literature. After discussing briefly the previous results, we will present recent results of the quantitative observability result of the 1D Schrödinger equation on real line. Our proof relies on different techniques for low-frequency and high-frequency estimates. This talk is based on the joint work with Pei Su and Chenmin Sun.
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