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Cung The Anh,Vu Manh Toi,Phan Thi Tuyet 대한수학회 2024 대한수학회지 Vol.61 No.2
This paper studies the existence of weak solutions and the stability of stationary solutions to stochastic 3D globally modified Navier-Stokes equations with unbounded delays in the phase space $BCL_{-\infty}(H)$. We first prove the existence and uniqueness of weak solutions by using the classical technique of Galerkin approximations. Then we study stability properties of stationary solutions by using several approach methods. In the case of proportional delays, some sufficient conditions ensuring the polynomial stability in both mean square and almost sure senses will be provided.
Global attractors for nonlocal parabolic equations with a new class of nonlinearities
Cung The Anh,Le Tran Tinh,Vu Manh Toi 대한수학회 2018 대한수학회지 Vol.55 No.3
In this paper we consider a class of nonlocal parabolic equations in bounded domains with Dirichlet boundary conditions and a new class of nonlinearities. We first prove the existence and uniqueness of weak solutions by using the compactness method. Then we study the existence and fractal dimension estimates of the global attractor for the continuous semigroup generated by the problem. We also prove the existence of stationary solutions and give a sufficient condition for the uniqueness and global exponential stability of the stationary solution. The main novelty of the obtained results is that no restriction is imposed on the upper growth of the nonlinearities.
GLOBAL ATTRACTORS FOR NONLOCAL PARABOLIC EQUATIONS WITH A NEW CLASS OF NONLINEARITIES
Anh, Cung The,Tinh, Le Tran,Toi, Vu Manh Korean Mathematical Society 2018 대한수학회지 Vol.55 No.3
In this paper we consider a class of nonlocal parabolic equations in bounded domains with Dirichlet boundary conditions and a new class of nonlinearities. We first prove the existence and uniqueness of weak solutions by using the compactness method. Then we study the existence and fractal dimension estimates of the global attractor for the continuous semigroup generated by the problem. We also prove the existence of stationary solutions and give a sufficient condition for the uniqueness and global exponential stability of the stationary solution. The main novelty of the obtained results is that no restriction is imposed on the upper growth of the nonlinearities.