Strong \begin{document}$ (L^2,L^\gamma\cap H_0^1) $\end{document} -continuity in initial data of nonlinear reaction-diffusion equation in any space dimension

In this paper we study the continuity in initial data of a classical reaction-diffusion equation with arbitrary \begin{document}$ p>2 $\end{document} order nonlinearity and in any space dimension \begin{document}$ N \geqslant 1 $\end{document} . It is proved that the weak solutions can be \begin{document}$ (L^2, L^\gamma\cap H_0^1) $\end{document} -continuous in initial data for arbitrarily large \begin{document}$ \gamma \geqslant 2 $\end{document} (independent of the physical parameters of the system), i.e., can converge in the norm of any \begin{document}$ L^\gamma\cap H_0^1 $\end{document} as the corresponding initial values converge in \begin{document}$ L^2 $\end{document} . In fact, the system is shown to be \begin{document}$ (L^2, L^\gamma\cap H_0^1) $\end{document} -smoothing in a H \begin{document}$ \ddot{\rm o} $\end{document} lder way. Applying this to the global attractor we find that, with external forcing only in \begin{document}$ L^2 $\end{document} , the attractor \begin{document}$ \mathscr{A} $\end{document} attracts bounded subsets of \begin{document}$ L^2 $\end{document} in the norm of any \begin{document}$ L^\gamma\cap H_0^1 $\end{document} , and that every translation set \begin{document}$ \mathscr{A}-z_0 $\end{document} of \begin{document}$ \mathscr{A} $\end{document} for any \begin{document}$ z_0\in \mathscr{A} $\end{document} is a finite dimensional compact subset of \begin{document}$ L^\gamma\cap H_0^1 $\end{document} . The main technique we employ is a combination of a Moser iteration and a decomposition of the nonlinearity, by which the interpolation inequalities are avoided and the new continuity result is obtained without any restrictions on the order \begin{document}$ p>2 $\end{document} of the nonlinearity and the space dimension \begin{document}$ N \geqslant 1 $\end{document} .