Is there any relativistic constraint on the rate of change of a scalar field?

Consider a scalar field $phi(t, x, y, z)$ obeying the waves equation with an Higgs-like potential (the "mexican hat"):
$$tag{1}
mathcal{V}(phi) = frac{lambda}{4} (phi^2 – phi_0^2)^2,
$$
where $lambda > 0$ and $phi_0$ are constants. $phi_0$ is the value of the "true vacuum" field, which can be positive or negative. The waves equation of motion of the self-interacting field is this (I’m using units such that $c equiv 1$):
$$tag{2}
frac{partial^2 phi}{partial t^2} – frac{partial^2 phi}{partial x^2} – frac{partial^2 phi}{partial y^2} – frac{partial^2 phi}{partial z^2} + lambda (phi^2 – phi_0^2) , phi = 0.
$$
To solve the waves equation (2), we need a consistent set of initial conditions and boundary conditions. For the initial conditions, we usually give the field values at time $t = 0$ and its rate of change:
begin{align}
phi(0, x, y, z) &= mathcal{F}(x, y, z), tag{3} [2ex]
frac{partial phi}{partial t} , bigg|_{t = 0} &= mathcal{G}(x, y, z). tag{4}
end{align}
As far as I know, there is no constraint on the values of $phi(0, x, y, z)$, in the classical theory of relativistic fields. The function $mathcal{F}$ can be anything.

But is there any constraint on the values of the field derivative? Can the function $mathcal{G}$ be completely arbitrary too in relativity? (this time derivative is not the same as a real velocity, which of course is constrained by causality: $v < 1$).