I am trying to use Newton's method to find the stationary solutions of the integro-differential equation of the form $$\frac{\partial u(r,t)}{\partial t} = -u(r,t) + \int_{\mathbb{R}^{2}}w(r - r')f(u(r',t))\mathrm{d}r' \quad\quad r\in\mathbb{R}^{2}. $$ I began by implementing Newton's method for the 1D version of the above, where $u(r)$ is discretized on the domain to give a vector in $\mathbb{R}^{n}$. Newtons method then amounts to solving at each iteration the system of equations $$ Jx = -F,$$ where $J$ is the Jacobian, $J\in\mathbb{R}^{n\times n}$, $x\in\mathbb{R}^{n}$ is the current iterate, and $F\in\mathbb{R}^{n}$ is the discretized right-hand side of the first equation. I implemented this just fine in this 1D case, using GMRES to solve the system of equations at each step.
I must now attempt to do the same procedure for the 2D case, where $r\in\mathbb{R}^{2}$ and therefore $u(r)$ is now a discretized surface function - i.e. an $n\times n$ matrix. But I don't know how to proceed. It seems to me that at each iteration of Newton, I would need to solve something like $$ Jx = -F\quad\quad x,F\in\mathbb{R}^{n\times n}.$$ Does this mean that I should essentially solve the above system one row or one column at a time perhaps?
I'm not quite sure if I've made my situation clear. To give some intuition, the 2D stationary states I'm looking for will look like this: 
So the above surface function is discretized to give a matrix $u\in\mathbb{R}^{n\times n}$. I can't seem to find any help online about how to do Newton's method on a surface like this! Any help would be much appreciated. Thanks.