The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction–diffusion equations

Abstract

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 ( Ω ) , H 0 1 ( Ω ) and H 2 ( Ω ) ∩ H 0 1 ( Ω ) , 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 ( Ω ) , H 0 1 ( Ω ) and H 2 ( Ω ) ∩ H 0 1 ( Ω ) , the results in this part are new and appear to be optimal.