If $u$ satisfies the 1D heat equation, show that $u^2$ satisfies another PDE

Question

Consider heat conduction in a rod described by $$ \frac{\partial u}{\partial t} = \kappa \frac{\partial^2 u}{\partial x^2},$$ with constant thermal conductivity $\kappa$. Show that if $u$ satisfies this equation, $u^2$ satisfies $$\frac{\partial u^2}{\partial t} = 2\kappa\left[\frac{\partial}{\partial x}\left(u\frac{\partial u}{\partial x}\right)-\left(\frac{\partial u}{\partial x}\right)^2\right].$$

I have no clue how to show the above. Should I use the chain rule and product rule somehow, or is it something else entirely? Help would be greatly appreciated.

Answer 1

Chain rule of course. Assume $\partial_t u = \kappa \partial_{xx}u$: $$ \partial_t (u^2) = 2u\partial_t u = 2u \kappa \partial_{xx}u $$ On the other hand you need the product rule: $$ 2\kappa \left(\partial_x (u\partial_x u) - (\partial_x u)^2 \right) = 2\kappa ((\partial_x u)^2 + u \partial_{xx}u - (\partial_x u)^2) = 2u\kappa \partial_{xx}u $$

This means that $u^2$ indeed satisfies the desired equation.