We consider the Fix-Caginalp equation with the Neumann boundary
condition in ${\bf R}^n$ with $n=1,2,3$.
We obtain a global solution by the existence of the Lyapunov
function.
After, we construct a dynamical system corresponding to the equation.
By the existence of the Lyapunov function,
the $\omega$-limit set is included in the set of its stationary
solution.
We treat its dynamical properties such as a global attractor,
absorbing set, exponential attractor and so on.
It is important to obtain the estimate independent of the initial
value.
Finally, we construct an exponential attractor.
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