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UNIFORM ATTRACTORS FOR NON-AUTONOMOUS NONCLASSICAL DIFFUSION EQUATIONS ON ℝ<sup>N</sup>
Anh, Cung The,Nguyen, Duong Toan Korean Mathematical Society 2014 대한수학회보 Vol.51 No.5
We prove the existence of uniform attractors $\mathcal{A}_{\varepsilon}$ in the space $H^1(\mathbb{R}^N){\cap}L^p(\mathbb{R}^N)$ for the following non-autonomous nonclassical diffusion equations on $\mathbb{R}^N$, $$u_t-{\varepsilon}{\Delta}u_t-{\Delta}u+f(x,u)+{\lambda}u=g(x,t),\;{\varepsilon}{\in}(0,1]$$. The upper semicontinuity of the uniform attractors $\{\mathcal{A}_{\varepsilon}\}_{{\varepsilon}{\in}[0,1]}$ at ${\varepsilon}=0$ is also studied.
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.
Cung The Anh,Bui Huy Bach 대한수학회 2021 대한수학회지 Vol.58 No.1
We study a continuous data assimilation algorithm for the three-dimensional simplified Bardina model utilizing measurements of only two components of the velocity field. Under suitable conditions on the relaxation (nudging) parameter and the spatial mesh resolution, we obtain an asymptotic in time estimate of the difference between the approximating solution and the unknown reference solution corresponding to the measurements, in an appropriate norm, which shows exponential convergence up to zero.
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.
LONG-TIME BEHAVIOR FOR SEMILINEAR DEGENERATE PARABOLIC EQUATIONS ON ℝ<sup>N</sup>
Cung, The Anh,Le, Thi Thuy Korean Mathematical Society 2013 대한수학회논문집 Vol.28 No.4
We study the existence and long-time behavior of solutions to the following semilinear degenerate parabolic equation on $\mathbb{R}^N$: $$\frac{{\partial}u}{{\partial}t}-div({\sigma}(x){\nabla}u+{\lambda}u+f(u)=g(x)$$, under a new condition concerning a variable non-negative diffusivity ${\sigma}({\cdot})$. Some essential difficulty caused by the lack of compactness of Sobolev embeddings is overcome here by exploiting the tail-estimates method.
EXISTENCE AND LONG-TIME BEHAVIOR OF SOLUTIONS TO NAVIER-STOKES-VOIGT EQUATIONS WITH INFINITE DELAY
Anh, Cung The,Thanh, Dang Thi Phuong Korean Mathematical Society 2018 대한수학회보 Vol.55 No.2
In this paper we study the first initial boundary value problem for the 3D Navier-Stokes-Voigt equations with infinite delay. First, we prove the existence and uniqueness of weak solutions to the problem by combining the Galerkin method and the energy method. Then we prove the existence of a compact global attractor for the continuous semigroup associated to the problem. Finally, we study the existence and exponential stability of stationary solutions.
LONG-TIME BEHAVIOR OF A FAMILY OF INCOMPRESSIBLE THREE-DIMENSIONAL LERAY-α-LIKE MODELS
Anh, Cung The,Thuy, Le Thi,Tinh, Le Tran Korean Mathematical Society 2021 대한수학회보 Vol.58 No.5
We study the long-term dynamics for a family of incompressible three-dimensional Leray-α-like models that employ the spectral fractional Laplacian operators. This family of equations interpolates between incompressible hyperviscous Navier-Stokes equations and the Leray-α model when varying two nonnegative parameters 𝜃<sub>1</sub> and 𝜃<sub>2</sub>. We prove the existence of a finite-dimensional global attractor for the continuous semigroup associated to these models. We also show that an operator which projects the weak solution of Leray-α-like models into a finite-dimensional space is determining if it annihilates the difference of two "nearby" weak solutions asymptotically, and if it satisfies an approximation inequality.
UNIFORM ATTRACTORS FOR NON-AUTONOMOUS NONCLASSICAL DIFFUSION EQUATIONS ON RN
Cung The Anh,Nguyen Duong Toan 대한수학회 2014 대한수학회보 Vol.51 No.5
We prove the existence of uniform attractors A" in the space H1(RN)∩Lp(RN) for the following non-autonomous nonclassical diffusion equations on RN, ut − ε△ut −△u + f(x, u) + u = g(x, t), ε∈ (0, 1]. The upper semicontinuity of the uniform attractors {Aε}ε∈[0,1] at ε = 0 is also studied.
Existence and long-time behavior of solutions to Navier-Stokes-Voigt equations with infinite delay
Cung The Anh,Dang Thi Phuong Thanh 대한수학회 2018 대한수학회보 Vol.55 No.2
In this paper we study the first initial boundary value problem for the 3D Navier-Stokes-Voigt equations with infinite delay. First, we prove the existence and uniqueness of weak solutions to the problem by combining the Galerkin method and the energy method. Then we prove the existence of a compact global attractor for the continuous semigroup associated to the problem. Finally, we study the existence and exponential stability of stationary solutions.
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.