Weak convergence of attractors of reaction–diffusion systems with randomly oscillating coefficients

ABSTRACT

We consider reaction–diffusion systems with random rapidly oscillating coefficient. We do not assume any Lipschitz condition for the nonlinear function in the system, so, the uniqueness theorem for the corresponding initial-value problem may not hold for the considered reaction–diffusion system. Under the assumption that the random function is ergodic and statistically homogeneous in space variables we prove that the trajectory attractors of these systems tend in a weak sense to the trajectory attractors of the homogenized reaction–diffusion systems whose coefficient is the average of the corresponding term of the original systems.

KEYWORDS:

• Attractors
• random homogenization
• reaction–diffusion systems
• nonlinear equations
• weak convergence

AMS SUBJECT CLASSIFICATIONS:

• 35B40
• 35B41
• 35B45
• 35Q30

Notes

No potential conflict of interest was reported by the authors.

To the blessed memory of V. V. Zhikov.

1 The image of the attractor of evolutionary equation was taken from the internet https://fr.wikipedia.org/wiki/Attracteur.

Additional information

Funding

Work of GAC and VVC is partially supported by the Russian Foundation of Basic Researches [projects 18-01-00046], [17-01-00515]. Work of KAB is supported in part by the Committee of Science of the Ministry of Education and Science of the Republic of Kazakhstan (CS MES RK) by [grant number AP05131707].