In this article, the well-posedness and long-time behavior of a nonclassical diffusion equation of Kirchhoff type are considered. Using the method of Galerkin approximation, the existence and uniqueness of solutions a...In this article, the well-posedness and long-time behavior of a nonclassical diffusion equation of Kirchhoff type are considered. Using the method of Galerkin approximation, the existence and uniqueness of solutions are proved. At last, the existence of global attractors and its upper semicontinuous property are discussed.展开更多
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 H<sup>1...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 H<sup>1</sup>(R<sup>n</sup>). 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.展开更多
In this paper,we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x)∈L^(2)...In this paper,we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x)∈L^(2)(Ω).In the framework of time-dependent spaces,we verify the existence of absorbing sets and the asymptotic compactness of the process,then we obtain the existence of the time-dependent global attractor A={A_t}t∈Rin Mt.Furthermore,we achieve the regularity of A,that is,A_(t) is bounded in M_(t)^(1) with a bound independent of t.展开更多
We prove the existence of global attractors in H0^1 （Ω） for a nonclassical diffusion equation. Two types of nonlinearity f are considered： one is the critical exponent, and the other is the polynomial growth of ar...We prove the existence of global attractors in H0^1 （Ω） for a nonclassical diffusion equation. Two types of nonlinearity f are considered： one is the critical exponent, and the other is the polynomial growth of arbitrary order.展开更多
In this paper, we consider the stochastic nonclassical diffusion equation with fading memory on a bounded domain. By decomposition of the solution operator, we give the necessary condition of asymptotic smoothness of ...In this paper, we consider the stochastic nonclassical diffusion equation with fading memory on a bounded domain. By decomposition of the solution operator, we give the necessary condition of asymptotic smoothness of the solution to the initial boundary value problem, and then we prove the existence of a random attractor in the space M1=D（A2^-1）×Lu^2（R＋（A2^-1））,where A = --A with Dirichlet boundary condition.展开更多
In this paper,we first survey existed theorems and propose all 46 related open problems of the existence of global attractors for autonomous dynamical systems,then establish a new existence theorem of global attractor...In this paper,we first survey existed theorems and propose all 46 related open problems of the existence of global attractors for autonomous dynamical systems,then establish a new existence theorem of global attractors which will be applied to a nonclassical diffusion equation for the norm-to-weak continuous,weakly compact semigroup on H_(0)^(1)(Ω)and H^(2)(Ω)∩H_(0)^(1)(Ω)respectively.As an application of this new existence theorem of global attractors,we obtain the existence of the global attractors on H_(0)^(1)(Ω)and H^(2)(Ω)∩H_(0)^(1)(Ω)respectively for a nonclassical-diffusion equation.展开更多
基金National Natural Science Foundation of China ( No. 11031003) Fund of Excellent Young Teachers in Shanghai,China( No.shgcjs008) Initial Fund of Shanghai University of Engineering Science,China( No. A-0501-11-016)
文摘In this article, the well-posedness and long-time behavior of a nonclassical diffusion equation of Kirchhoff type are considered. Using the method of Galerkin approximation, the existence and uniqueness of solutions are proved. At last, the existence of global attractors and its upper semicontinuous property are discussed.
文摘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 H<sup>1</sup>(R<sup>n</sup>). 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.
基金supported by the National Natural Science Foundation of China(No.12171082)the Fundamental Research Funds for the Central Universities(No.2232023G-13).
文摘In this paper,we discuss the long-time behavior of solutions to the nonclassical diffusion equation with fading memory when the nonlinear term f satisfies critical exponential growth and the external force g(x)∈L^(2)(Ω).In the framework of time-dependent spaces,we verify the existence of absorbing sets and the asymptotic compactness of the process,then we obtain the existence of the time-dependent global attractor A={A_t}t∈Rin Mt.Furthermore,we achieve the regularity of A,that is,A_(t) is bounded in M_(t)^(1) with a bound independent of t.
基金Supported in part by the NSFC,Grant(10471056)Trans-Century Training Programme Foundation for the Talents of the State Education Commission
文摘We prove the existence of global attractors in H0^1 （Ω） for a nonclassical diffusion equation. Two types of nonlinearity f are considered： one is the critical exponent, and the other is the polynomial growth of arbitrary order.
文摘In this paper, we consider the stochastic nonclassical diffusion equation with fading memory on a bounded domain. By decomposition of the solution operator, we give the necessary condition of asymptotic smoothness of the solution to the initial boundary value problem, and then we prove the existence of a random attractor in the space M1=D（A2^-1）×Lu^2（R＋（A2^-1））,where A = --A with Dirichlet boundary condition.
基金Supported by the NNSF of China(Grant No.12171082)The fundamental research funds for the central universities(Grant Nos.2232022G-13 and 2232023G-13)by a grant from Science and Technology Commission of Shanghai Municipality。
文摘In this paper,we first survey existed theorems and propose all 46 related open problems of the existence of global attractors for autonomous dynamical systems,then establish a new existence theorem of global attractors which will be applied to a nonclassical diffusion equation for the norm-to-weak continuous,weakly compact semigroup on H_(0)^(1)(Ω)and H^(2)(Ω)∩H_(0)^(1)(Ω)respectively.As an application of this new existence theorem of global attractors,we obtain the existence of the global attractors on H_(0)^(1)(Ω)and H^(2)(Ω)∩H_(0)^(1)(Ω)respectively for a nonclassical-diffusion equation.
