Hello!!!

Suppose that $u(t,x)$ is a solution of the heat equation $u_t-\Delta u=0$ in $(0,+\infty) \times \mathbb{R}^n$. I want to show that $u_k \equiv u(k^2 t, kx)$ is also a solution of the heat equation in $(0,+\infty) \times \mathbb{R}^n, \forall x \in \mathbb{R}^n$.

If we have a function $u(g,f)$ then the derivative in respect to $t$ is $\frac{\partial{u}}{\partial{g}} \frac{dg}{dt}+\frac{\partial{u}}{\partial{f}} \frac{df}{dt}$, right?

But then we would get that $\frac{\partial{u_k}}{\partial{t}}=k^2 \frac{\partial{u}}{\partial{k^2 t}}$.

But does the derivative $\frac{\partial{u}}{\partial{k^2 t}}$ make sense?

Or haven't I applied correctly the chain rule?