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.

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 .

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Based on a new a priori estimate method, so-called asymptotic a priori estimate, the existence of a global attractor is proved for the wave equation u tt + kg(u t ) − Δu + f (u) = 0 on a bounded domain Ω ⊂ R 3 with Dirichlet boundary conditions. The nonlinear damping term g is supposed to satisfy the growth condition C 1 (|s| − C 2 ) |g(s)| C 3 (1 + |s| p ), where 1 p < 5; the damping parameter k (> 0) is arbitrary; the nonlinear term f is supposed to satisfy the growth condition |f (s)| C 4 (1 + |s| q ), where q 2. It is remarkable that when 2 < p < 5, we positively answer an open problem in Chueshov and Lasiecka [I. Chueshov, I. Lasiecka, Long-time behavior of second evolution equations with nonlinear damping, Math. Scuola Norm. Sup. (2004)] and improve the corresponding results in Feireisl [E. Feireisl, Global attractors for damped wave equations with supercritical exponent, J. Differential Equations 116 (1995) 431-447].

It is shown that a stochastic partial differential equation of the reaction–diffusion type on time-varying domains obtained by a temporally continuous dependent spatially diffeomorphic transformation of a reference domain, which is bounded with a smooth boundary, generates a "partial-random" dynamical system, which has a pathwise nonautonomous pullback attractor.