Suppose that $v(t, z)$ is analytic with respect to the complex variable $z$ and differentiable with respect to the real variable $t$ and satisfies the partial differential equation $$\frac{\partial v}{\partial t}(t, z)+\frac{\partial^{2} v}{\partial z^{2}}(t, z)=0,$$ for all $t>0$ and $z=x+iy$, $x\in \mathbb{R}$, $|y|\le r$.

Then, what partial differential equation with respect to the real variables $t>0$ and $x\in \mathbb{R}$ satisfies $v(t, x)$ ?

(It is obtained simply by replacing the complex $z$ by the real $x$ in the original equation, and if this is the answer, how could be proved that ?)

Thank you very much in advance.