In Aspects of Symmetry, Coleman says (p. 185)
''Most of the simple field theories with which we are familiar have the property that all of their non-singular solutions of finite total energy are dissipative. This is the case for Maxwell's equations, the Klein-Gordon equation, etc.''
However, isn't it the case that the mentioned equations for regular initial data are dispersive? For example, we take the vacuum Maxwell equations - they will have the form (for the electric/magnetic field)
\begin{equation} y_{xx}-c^{-2}y_{tt}=0. \end{equation}
If we take $y(x,0)=\sin x$ and $\partial_t y(x,0)=(1/\epsilon)\sin x$ for the electric field and mutatis mutandis for the magnetic field, we obtain the electromagnetic wave - a dispersive solution.