I would like to linearise the following differential operator (which is the McKendrik-von Forster equation combined with a nonlocal nonlinearity):
\begin{align} \dfrac{\partial n(t, x)}{\partial t} + \dfrac{\partial g\big(I(t), x\big)n(t, x)}{\partial x} &= \mu(I(t), x)n(t, x) && t > 0, x \in \Omega \subset R, \\ I(t) &= \int_{\Omega} \gamma(x)n(t, x) \, dx, \end{align}
where the first equation is the advection at speed $ g $ of individuals of size $ x $ at time $ t $, $ \mu $ is a loss rate, and $ I $ is the nonlocal nonlinearity. The function $ \gamma $ represent a weight, i.e., the influence of a particular $ n(x, \cdot) $ on the rates $ g $ and $ \mu $.
I would like to linearise this operator around a solution $ \tilde{n} $, but I am not sure how to start. If I rewrite $ g $ as a function of $ n $ only, then I can follow what is done on this page: How do you "linearize" a differential operator to get its symbol?. However, I have the feeling that this is not correct to do it this way because the nonlinearity is nonlocal.
Should I instead compute $ I $ for a perturbed solution ($ \varepsilon $ as small as I wish): $$ n = \tilde{n} + \varepsilon v $$ and therefore \begin{align} I(t) &= \int \gamma \tilde{n} \, dx + \varepsilon \int \gamma v \, dx \\ &= \tilde{I}(t) + \varepsilon I_v(t) \end{align} and then develop $ g $ around $ \tilde{I} $ as: $$ g\big(I(t), x\big) = g\big(\tilde I(t), x \big) + \varepsilon I_v(t) \partial_I g(\tilde I, x) $$ and then continue the calculus with the differential operator keeping only terms of order 0 and 1, and dropping all the $ \varepsilon^2 $ terms? I guess at the end I will get the same equation with $ v $ instead of $ n $, with few $ \tilde n $, and this equation should be linear in terms of $ v $. So can anyone confirm which path I should take and if what I am thinking is correct (I assume that the functions are all nice, I am not asking about the existence of a solution).
