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Generalization of Propagation Speed with Variation Structure in Reaction-diffusion System of Gradient Type

Yi-Yun Lee (National Taiwan University)

Abstract:

We study the following initial value problem :

, (1)

where

,

is a bounded domain with smooth boundary,

is in

, with either Neumann or Dirichlet boundary conditions. This problem describes a reaction-diffusion system with equal diffusion coefficients defined inside an n-dimensional cylinder

.

We assume that

, so

is a trivial solution of above equations.

If there exists a function

such that

,

then we called it reaction-diffusion of gradient type.

Traveling solutions are important in these systems and a lot of researchers study related topics, including existence and stability of traveling wave solutions in vary domains and traveling wave solutions selections.

Furthermore, following from ideas in Muratov's study [1], we introduce "wave-like" solutions and propagation speed of such solutions by variation structures, and we proved that propagation speed of wave-like solutions is monotone increasing.

References:

[1] CB Muratov. A global variational structure and propagation of disturbances in reaction-diffusion systems of gradient type. *Discrete & Continuous Dynamical Systems-B*, 4(4):867, 2004.