A new concept, called the norm-to-weak continuous semigroup in Banach space is introduced. The definition is the following: Definition: Let be a Banach space and be a family of operators on . We say that is a norm-to-weak continuous semigroup on , if satisfies that , , if and in .
In evolution equation, this type of semigroup corresponds to the solution that only satisfies weaker stability, and generally, it is neither continuous (i.e. norm-to-norm) nor weak continuous (i.e., weak-to-weak). But continuous semigroups and the weak continuous semigroups are norm-to-weak continuous semigroups.
A technical method to verify which semigroup is norm-to-weak continuous is given. A general method which gives a necessary and sufficient condition for the existence of the global attractor for this kind of semigroup is established, too. As an application of theoretical results, 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 is obtained. The global attractors are obtained in and for the case where the external forcing term , and in and for the case where .