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I want to ask a question about the eigenvalue problem (EVP) and the initial value problem (IVP).

Let's say we are solving this linear equation [itex]\frac{\partial u}{\partial t}=\mathcal{L}u[/itex], the operator L is dependent on some parameters like Reynolds number.

I first check the eigenvalue of the equation at a specific parameter set.

My question is if the eigenvalue at this specific parameter is positive (meaning the system is unstable), now I solved the IVP counterpart [itex]\frac{u^{n+1}-u^n}{dt}=\mathcal{L}u[/itex] with an ARBITRARY initial condition, will this initial condition amplifies over all the time history (especially at the early phase, because I know asymptotically the system is governed by the positive eigenvalue)?

Thanks in advance

Jo