Diffusion and dispersion relation

I’m looking at some dispersion relations for some complex systems and realised I actually don’t have a clear understanding of what physics I can get from a dispersion relation from equations that produce complex relations between frequencies and wavevectors. For instance, if you take a diffusion equation

$$dot u=nabla^2u$$

for an initial condition $u_0(x)=u(x,t=0)$, then this equation can be simply solved in Fourier space, and the solution is

$$u(x,t) = int frac{d^dq}{(2pi)^d}tilde u_0(q)e^{-q^2t}e^{i qcdot r}$$

If the initial equation is transformed to Fourier space also in time we get a complex relation between frequency $omega$ and wavevector $k$,
$$iomega+q^2=0.$$

Which will be generaly the case for a scalar field $u$ for any overdamped dynamics $dot u= F[u]$ for any $F$.

My question is what do we make of a relation of that kind? I’m guessing something along the lines of, since this is overdamped and just a scalar field, you don’t expect wave-like solutions, so since an initial condition won’t propagate like a wave, there’s no real correspondence between $omega $ and $q$. I know this is extremely vague, but I’m looking for some physical insight on this.

To expand the question a bit, if $F[u]$, for instance $F[u]=nabla^2u +A[u]$ for a nonlinear operator $A$, we can ‘solve’ for $u$ through its linear propagator
$$tilde u = frac{1}{iomega+q^2}tilde A[tilde u]$$

How do we interpret this propagator function physically?

Thanks!