Is there any partial differential equation such that the its solution evolves as partial Fourier integral (continuous version of partial sum) of a function $f(x)$ which might be an condition or something of that sort? or the condition might be $F(\omega)$.
Clarification
Basically the solution should be $$u(x,\omega) = \int_0^{\omega}[A(\omega)\\cos\omega x + B(\omega)\sin\omega x]d\omega$$ where $F(\omega) = A(\omega)+iB(\omega)$ is the Fourier transform of a function $f(x)$.
Here $u(.,\omega)$ evolves with $\omega$.
Some work :
It seems $$u_{xx\omega} = -\omega^2u_{\omega}$$
and unlike usual pde's the condition is at $\omega = \infty$ rather than at $\omega = 0$
(Converted and fixed from a comment)
Let $F(t,x) = \rho(t) \exp(itx)$, then the solution to the PDE $$ \partial_t u = F $$ with $0$ initial data is exactly $$ u(t,x) = \int_0^t \rho(s) \exp(isx) \mathrm{d}s. $$