Propagation speeds of disturbances are bounded for a class of reaction–diffusion systems

Abstract

It is shown in a rigorous way that propagation speeds of disturbances are bounded for a class of reaction–diffusion systems. It turns out that solutions for various initial states are confined by traveling waves. A new technique is developed for the construction of the comparison functions. The technique is based on the operator-splitting methodology, which is known as a numerical computation method. By using an exact solution of the Fisher equation we can make a simple proof. The upper bounds of the speeds are given explicitly in terms of the diffusion constants and the Lipschitz norms of the reaction terms.

Keywords:

• Reaction–diffusion system
• Comparison technique
• Operator-splitting method
• Traveling wave
• 2000 Mathematics Subject Classifications: 35B05
• 35K45
• 35K57

Acknowledgments

This work is supported by Kyushu University 21st Century COE Program, Development of Dynamic Mathematics with High Functionality, of the Ministry of Education, Culture, Sports, Science and Technology of Japan. The author thanks Prof. Shimozawa for his stimulus without which, the research would not have been completed. He is also grateful to Prof. Maruno for informing him of the exact solution.