Suppose $F:C^1(\Omega, [0,T]) \to C^1(\Omega, [0,T])$ with $F(u) = u_t - f(x, t, u, u_x).$
How do I calculate the Frechet derivative of $F$ at the point $w = f(x,t, 0, 0)t$?
It should be $F'(w, v) = v_t - \frac{\partial f}{\partial z}\bigg|_{w}v - \frac{\partial f}{\partial p}\bigg|_{w}v_x$ apparently.
Maybe another day I can do this but forming the difference and then considering another difference to get out the partial derivative is confusing me!
Thanks