Homogenization of integro-differential equation of Burgers type

Abstract

Non-linear partial differential equation of Burgers type with integral term is considered. Its coefficients are rapidly oscillating functions. The equation describes wave propagation in stratified media with relaxation. Existence and uniqueness of solution is proved. Homogenized equation with constant coefficients is constructed. Convergence of exact solution to homogenized problem solution is proved.

Keywords:

• nonlinear equation
• relaxation
• energy estimates
• Galerkin method
• homogenization
• asymptotic expansion

AMS Subject Classifications: :

• 35B27
• 35Q53
• 35C20

Acknowledgements

The study was supported by the French Ministry of Foreign Affairs Program–Eiffel Doctorate Scholarship (Programme Eiffel Doctorat), the multi-disciplinary PPF grant ALLIANA of the Ministery of Research of France. The authors are greatful to Professors Andrey Amosov and Ilya Kostin for useful remarks and suggestions, to Professors Evgeny Lapshin and Oleg Rudenko for numerious discussions of the setting of the problem.