Let $\Omega \subset \mathbb{R}^{2}$ be an open set such that $\mathbf{0} \in \Omega$. Let $A := \Omega \setminus (\{0\}\times \mathbb{R})$, that is, $A$ is $\Omega$ with the $y$-axis removed.
Let $F: (A\cup \{\mathbf{0}\})\times\mathbb{R}^5 \to \mathbb{R}$, such that $F$ is real analytic on the open set $A \times \mathbb{R}^5$. Let us consider the partial differential equation on the set $A$
$$ \frac{\partial^2 \varphi}{\partial y^2} = F(x,y,\varphi, \frac{\partial \varphi}{\partial x}, \frac{\partial \varphi}{\partial y}, \frac{\partial^2\varphi}{\partial x\partial y}, \frac{\partial^2\varphi}{\partial x^2}) $$
with the initial conditions
$$ \varphi(x,0) = \frac{\partial \varphi}{\partial y}(x,0) = 0 $$
By Cauchy-Kovalevskaya this PDE has a solution in an open neighborhood of the $x$-axis inside the set $A$. Assume that the solution exists in all of $A$ and $F:D\rightarrow \mathbb{R}$, where $D = A \times \mathbb{R}^{5} \subset \mathbb{R}^{7}$. Furthermore assume that $F$ exists when $x=0$ and $y=0$.
Under what circumstances is it possible to extend this solution to a full neighbourhood of $\mathbf{0}$ ?
