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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