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．Seminars

	Nonlinear Phenomena in Evolutionary Partial Differential Equations

15:30 - 17:30, September 29, 2020 (Tuesday)
R440, Astronomy-Mathematics Building, NTU
(台灣大學天文數學館 440室)
Generalization of Propagation Speed in Reaction-diffusion Systems of Gradient Type
Yi-Yun Lee (National Taiwan University)

Abstract:

We study the following initial value problem:

$u_t=\Delta u+f(u),\ u(x,0)=u_0(x)$ , (1)

where $u=u(x,t)\in \mathbb{R}^m$ , $x\ \in \Sigma=\mathbb{R}\times \Omega,\ \Omega \subset \mathbb{R}^{n-1}$ is a bounded domain with smooth boundary, $f: \mathbb{R}^m \rightarrow \mathbb{R}^m$ is in $C^{\infty}$ , with either Neumann or Dirichlet boundary conditions. This problem describes a reaction-diffusion system with equal diffusion coefficients defined inside an n-dimensional cylinder $\Sigma$ .

We assume that $f(0) = 0$ , so $u = 0$ is a trivial solution of above equations.

If there exists a function $V (u)$ such that

$f = -\nabla_u V,\ V:\mathbb{R}^{m} \rightarrow \mathbb{R}$

then we called it reaction-diffusion system of gradient type.

Traveling solutions play an important role in such systems and a lot of researchers study related topics, including existence and stability of traveling wave solutions in vary domains and corresponding wave speed.

Furthermore, following from ideas in Muratov's study \cite{muratov2004global}, we introduce a wide class of solutions, called "wave-like" solutions, and show that well-defined instantaneous propagation speed for such solutions monotone converges to a limit at long time. Furthermore, with more assumptions, we could show it converges to traveling wave solution with such speed.