Assume that $u$ satisfies the heat equation: $$\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2}.$$
Then why does the conjugate of $u$ also satisfy the equation, i.e., $$\partial_t \bar{u}=\partial_x^2 \bar{u}.$$
This identity is used in the proof of the uniqueness of the solution given in Stein and Shakarchi's Fourier Analysis. I attached the excerpt below, but I don't know how to show that the equation holds for the conjugate as well. I would greatly appreciate any explanations.
