Could someone help me with the following problem:
Find all smooth functions u: R x [0, +infinity) ->R with the property that both u and u^n are solutions of the heat equation for some integer n>=2.
I do not know if this will help, but It may give you somewhere to start.
$\displaystyle u_t(x,t)=ku_{xx}(x,t)$
let $\displaystyle u(x,t) \mbox{ and } [u(x,t)]^n$ be solutions to the heat equation
let $\displaystyle z=[u(x,t)]^n$
$\displaystyle z_t=nu^{n-1}u_t$
$\displaystyle z_x=nu^{n-1}u_x$
$\displaystyle z_{xx}=n(n-1)u^{n-2}(u_x)^2+nu^{n-1}u_{xx}$
plugging the above into the heat eqaution gives
$\displaystyle z_t=kz_{xx}$
$\displaystyle nu^{n-1}u_t=k[n(n-1)u^{n-2}(u_x)^2+nu^{n-1}u_{xx}]$
$\displaystyle nu^{n-1}u_t=kn(n-1)u^{n-2}(u_x)^2+knu^{n-1}u_{xx}$
$\displaystyle nu^{n-1}u_t-knu^{n-1}u_{xx}=kn(n-1)u^{n-2}(u_x)^2$
$\displaystyle nu^{n-1}\underbrace{[u_t-ku_{xx}]}_{=0 \mbox{ because }u_t=ku_{xx}}=kn(n-1)u^{n-2}(u_x)^2$
$\displaystyle 0=kn(n-1)u^{n-2}(u_x)^2$
Now if $\displaystyle u(x,t)=X(x) \cdot T(t)$ we get
$\displaystyle 0=kn(n-1)[X(x)]^{n-2}[T(t)]^{n-2}[X_x(x)T(t)]^2$
$\displaystyle 0=kn(n-1)[X(x)]^{n-2}[T(t)]^{n}[X_x(x)]^2$
$\displaystyle 0=kn(n-1)[X(x)]^{n-2}[T(t)]^{n}[\frac{dX}{dx}]^2$
Maybe you can figure something out from here.
I hope this helps.
Good luck.