Assume we have the following Cauchy problem
$$\dot{u}=f(u,t,x),\; u(0,x)=\varphi(x).$$
Then if $f$ is continuous in all the variables, and has continuous partial derivatives $f_u$ and $f_x$, and if in addition we assume that $\varphi$ is continuously differentiable then the solution of the Cauchy problem is differentiable with respect to $x$ and the derivative solves the associated variational equation. See for instance here.
My question regards the differentiability of $f$ with respect to $u$, can we assume that it holds in a weak sense? Namely that $f$ is Lipschitz continuous with respect to $u$? Of course in such case I expect the solution to be weakly differentiable...
Thanks in advance.