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Seminars

Analysis Seminar

Uniform null-controllability of the Allen−Cahn equation linearized around transition layers

2026/04/28 (Tue.) 10:00~13:00
  • Date : 2026/04/28 (Tue.) 10:00~13:00
  • Location : online
  • Speaker : Kai Koike (Keio University)
  • Organizer : Tai-Ping Liu (AS)
  • Abstract : Consider the Allen−Cahn equation $u_t=\epsilon u_{xx}+(u-u^3)/\epsilon$ with a parameter $\epsilon>0$ posed on the interval $[0,1]$ with homogeneous Neumann boundary conditions. It is known that the equation possesses for each integer $n$, a unique (up to sign) steady solution $u_{n,\epsilon}$ with $n$ distinct zeros. Such solutions are called transition layers, and we study the uniform (with respect to $\epsilon$) null-controllability of the Allen−Cahn equation linearized around them, with distributed controls of the form $\chi_{\omega}(x)h(t,x)$, where $\chi_{\omega}$ is the indicator function of an open interval $\omega \subset (0,1)$. We show that if and only if the closure of $\omega$ contains all the zeros of $u_{n,\epsilon}$, the system is uniformly null-controllable in finite time. The proof is based on detailed spectral analysis and construction of bi-orthogonal families in time and space. This is a joint work with Vincent Laheurte (Université du Luxembourg) and Franck Sueur (Université du Luxembourg).

Click here to register: https://forms.gle/EhVyNa94MDEhQkqF9 (Deadline: 4pm, Apr. 27)
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