I'm taking a course in Non-linear physics and I have encountered the following Swift-Hohenberg equation in 2 dimensions:
$$\partial_t u = \mu u - (k_c + \nabla^2)^2 u -u^3$$
My teacher asked us to provide the boundary conditions on u for which it is possible to analyze the linear stability of the solution u=0 by looking for perturbations of the form $e^{st}e^{ikr}$ (where $r=(x,y)$). I'm sure it's very simple but it's has been a long time since I had to deal with this kind of stuff and I am a bit confused.
I know that the linearization of the equation is:
$$\partial_t u = \mu u - (k_c + \nabla^2)^2 u$$
And that plugging here the given perturbations I can find an expression for s in terms of $\mu$,$k_c$ and $k$.
So, I understand that this linear approximation is valid when $\vert u \vert << 1$ but I don't know how to quantify this using boundary conditions for $u$. How should I proceed?