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The existence of global attractors for the norm-to-weak continuous semigroup and application to the nonlinear reaction-diffusion equations. (English) Zbl 1101.35022

A new concept, called the norm-to-weak continuous semigroup in Banach space is introduced. The definition is the following: Definition: Let $X$ be a Banach space and ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ be a family of operators on $X$. We say that ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ is a norm-to-weak continuous semigroup on $X$, if ${\left\{S\left(t\right)\right\}}_{t\ge 0}$ satisfies that $S\left(0\right)=\text{Id}$, $S\left(t\right)S\left(s\right)=S\left(t+s\right)$, $S\left({t}_{n}\right){x}_{n}\to S\left(t\right)x$ if ${t}_{n}\to t$ and ${x}_{n}\to x$ in $X$.

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 ${L}^{p}\left({\Omega }\right)$ and ${H}_{0}^{1}\left({\Omega }\right)$ for the case where the external forcing term $g\in {H}^{-1}\left({\Omega }\right)$, and in ${L}^{2p-2}\left({\Omega }\right)$ and ${H}^{2}\left({\Omega }\right)\cap {H}_{0}^{1}\left({\Omega }\right)$ for the case where $g\in {L}^{2}\left({\Omega }\right)$.

##### MSC:
 35B41 Attractors (PDE) 47H20 Semigroups of nonlinear operators 35K57 Reaction-diffusion equations
##### Keywords:
norm-to-norm; weak-to-weak; polynomial growth nonlinearity