Author(s)

Fadlallah Mustafa Mosa¹, Abdelmajid Ali Dafallah^2*, Qiaozhen Ma³, Eshag Mohamed Ahmed⁴, Mohamed Y. A. Bakhet⁵

¹Department of Mathematics and Physics, Faculty of Education, University of Kassala, Kassala, Sudan.
²Department of Mathematics, Faculty of Petroleum and Hydrology Engineering, Alsalam University, Al-Fulah, Sudan.
³College of Mathematics and Statistics, Northwest Normal University, Lanzhou, China.
⁴Faculty of Pure and Applied Sciences, International University of Africa, Khartoum, Sudan.
⁵School of Mathematics, University of Juba, Juba, South Sudan.

This paper is concerned with the existence and upper semi-continuity of random attractors for the nonclassical diffusion equation with arbitrary polynomial growth nonlinearity and multiplicative noise in H¹(Rⁿ). First, we study the existence and uniqueness of solutions by a noise arising in a continuous random dynamical system and the asymptotic compactness is established by using uniform tail estimate technique, and then the existence of random attractors for the nonclassical diffusion equation with arbitrary polynomial growth nonlinearity. As a motivation of our results, we prove an existence and upper semi-continuity of random attractors with respect to the nonlinearity that enters the system together with the noise.

Random Attractors, Nonclassical Diffusion Equations, Asymptotic Compactness, Upper Semi-Continuity

Mosa, F. , Dafallah, A. , Ma, Q. , Ahmed, E. and Bakhet, M. (2022) Existence and Upper Semi-Continuity of Random Attractors for Nonclassical Diffusion Equation with Multiplicative Noise on Rⁿ. Journal of Applied Mathematics and Physics, 10, 3898-3919. doi: 10.4236/jamp.2022.1012257.