The 18th Prediction Science Seminar was held on June 6th (Fri.) 2023, from 10:30 to 12:00, and was presented by Prof. Masahiko Shimojo (Tokyo Metropolitan University).
  The main topic of the talk was the traveling wave solution to partial differential equations (PDEs) of the reaction-diffusion type. Professor Shimojo first introduced some PDE examples, such as the Allen-Cahn type and the KPP-Fisher type equations. The Allen-Cahn equation is bistable, and the KPP-Fisher equation is monostable. The talk then moved on to the spreading of the front of the solutions. He explained some methods to find the spreading speed. One approach is a particular solution-based approach, and the other focuses on the balance of the power of the fundamental solution.
  The main result introduced in the talk is the asymptotic analysis for spreading phenomena, where the solution with compactly supported initial data converges to a stable equilibrium, and the spreading front grows linearly. The key tool used was the Liouville-type theorem, which ensures that the entire solution is necessarily a constant solution. Professor Shimojo analyzed the spreading phenomenon with the Liouville-type theorem, Lyapunov functional, and scaling technique.
  He also introduced some applications of their method to non-local reaction-diffusion equations and the SIR model. The spreading speed for non-local reaction-diffusion equations is faster than the local one.