  In this paper, we study the continuity in initial data of a classical
reaction-diffusion equation with arbitrary $p>2$ order nonlinearity and in any
space dimension $N\geq 1$. It is proved that the weak solutions can be $(L^2,
L^\gamma\cap H_0^1)$-continuous in initial data for any $\gamma\geq 2$
(independent of the physical parameters of the system), i.e., can converge in
the norm of any $L^\gamma\cap H_0^1$ as the corresponding initial values
converge in $L^2$. Applying this to the global attractor we find that, with
external forcing only in $ L^2$, the attractor $\mathscr{A}$ attracts bounded
subsets of $L^2$ in the norm of any $L^\gamma\cap H_0^1$, and that every
translation set $\mathscr{A}-z_0$ of $\mathscr{A}$ for any $z_0 \in
\mathscr{A}$ is a finite dimensional compact subset of $L^\gamma\cap H_0^1$.
The main technique we employ is a combination of the mathematical induction and
a decomposition of the nonlinearity by which the continuity result is
strengthened to $(L^2, L^\gamma \cap H_0^1)$-continuity and, since
interpolation inequalities are avoided, the restriction on space dimension is
removed.
