In this paper, first, we introduce a new concept, called the norm-to-weak continuous semigroup in a Banach space, and give a technical theorem to verify this notion of continuity. Then we establish a general method which is necessary and sufficient to obtain the existence of the global attractor for this kind of semigroup. As an application, we obtain the existence of the global attractor for a nonlinear reaction-diffusion equation with a polynomial growth nonlinearity of arbitrary order and with some weak derivatives in the inhomogeneous term, the global attractors are obtained in L p ( ), H 1 0 ( ) and H 2 ( )∩H 1 0 ( ), respectively. A new a priori estimate method, called asymptotic a priori estimate, has been introduced. Since the solutions of the equation has no higher regularity and the semigroup associated the solutions is not continuous in L p ( ), H 1 0 ( ) and H 2 ( ) ∩ H 1 0 ( ), the results in this part are new and appear to be optimal.

The existence of a pullback attractor is established for the nonautonomous dynamical system generated by the weak solutions of a semilinear heat equation on time-varying domains with homogeneous Dirichlet boundary conditions. It is assumed that the spatial domains O t in R N are obtained from a bounded base domain O by a C 2 -diffeomorphism, which is continuously differentiable in the time variable, and are contained, in the past, in a common bounded domain.

We consider dynamical behavior of non-autonomous wave-type evolutionary equations with nonlinear damping, critical nonlinearity, and time-dependent external forcing which is translation bounded but not translation compact (i.e., external forcing is not necessarily time-periodic, quasi-periodic or almost periodic). A sufficient and necessary condition for the existence of uniform attractors is established using the concept of uniform asymptotic compactness. The required compactness for the existence of uniform attractors is then fulfilled by some new a priori estimates for concrete wave type equations arising from applications. The structure of uniform attractors is obtained by constructing a skew product flow on the extended phase space for the norm-to-weak continuous process.

The authors consider non-autonomous dynamical behavior of wave-type evolutionary equations with nonlinear damping and critical nonlinearity. These type of waves equations are formulated as non-autonomous dynamical systems (namely, cocycles). A sufficient and necessary condition for the existence of pullback attractors is established for norm-to-weak continuous non-autonomous dynamical systems, in terms of pullback asymptotic compactness or pullback κ−contraction criteria. A technical method for verifying pullback asymptotic compactness, via contractive functions, is devised. These results are then applied to the wave-type evolutionary equations with nonlinear damping and critical nonlinearity, to obtain the existence of pullback attractors. The required pullback asymptotic compactness for the existence of pullback attractors is fulfilled by some new a priori estimates for concrete wave type equations arising from applications. Moreover, the pullback κ−contraction criterion for the existence of pullback attractors is of independent interest.

We consider the dynamical behavior of the nonclassical diffusion equation with critical nonlinearity for both autonomous and nonautonomous cases. For the autonomous case, we obtain the existence of a global attractor when the forcing term only belongs to H −1 , this result simultaneously resolves a problem in Acta Mathematicae Applicatae Sinica 18 (2002), 273-276 related to the critical exponent. For the nonautonomous case, assumed that the time-dependent forcing term is translation bounded instead of translation compact, we first prove the asymptotic regularity of solutions, then the existence of a compact uniform attractor together with its structure and regularity has been obtained; finally, we show the existence of (nonautonomous) exponential attractors. 52 C. Sun and M. Yang / Dynamics of the nonclassical diffusion equationsfor all [t 1 , t 2 ] ⊂ R and for every v(x, s) ∈ L 2 (t 1 , t 2 ; L 2 (Ω)).We also need the following attraction transitivity lemma.Lemma 2.1 [23]. Let (M, d) be a metric space and U (t, τ ) be a Lipschitz continuous dynamical process in M, i.e.,for appropriate constants C and K which are independent of m i , τ and t. Assume further that there exist three subsets M 1 ,for some ν 1 , ν 2 > 0 and L 1 , L 2 > 0. Then it follows thatwhere ν = ν 1 ν 2 K+ν 1 +ν 2 and L = CL 1 + L 2 .