Consider for example, the Black Schole's equation $$ \partial_tu+0.5\sigma^2s^2\partial_{ss}u+rs\partial_su-ru=0 $$ on $[0,T]\times[0,\infty)$ subject to boundary conditions $u(s,T)=f(s)$.
The solution depends on the parameter $\sigma$, which is not a variable in the PDE, but it is possible to show that the solution is differentiable with respect to $\sigma$ for certain $f$ and $g$. Do there exists analysis tools and results which deal with smoothness of solutions with respect to perturbation of coefficients of PDEs?
In this particular case, the fundamental solution can be written down, and it is the transition kernel of the underlying geometric brownian motion. One can just differentiate the kernel, since $f$ has no dependence on $\sigma$. (Also okay if $f$ depends on $\sigma$ in a smooth way.)
However, if one has the same PDE on a domain with boundary such as $[0,T]\times[a,b]$ with $u(s,T)=f(s)$, $u(a,t)=h(t)$ and $u(b,t)=g(t)$, then it is still known that the PDE admits a unique solution, but it is unclear to me how to show this.
A similar question is asked here, but it is unclear to me how to apply the ODE theorem to the PDE case.
I have also found a paper which shows Frechet differentiability for parabolic PDE, but this does not translate to the classical differentiability definition here.
The paper is: http://www.sciencedirect.com/science/article/pii/S0895717707001859
It appears to me that the norm in this paper is not as strong as in the ODE case to give classic differentiablity. I have asked this question on math stack exchange, but the references in the comments are not directly linked to this question. In particular, the paper by Domanski says smooth dependence does not exist, but the assumptions for the problems are hugely different.
