Sign-changing self-similar solutions of the nonlinear heat equation with positive initial value

We consider the nonlinear heat equation $u_t-\Delta u=|u|^\alpha u$ on $\Bbb{R}^N$, where $\alpha>0$ and $N\ge 1$. We prove that in the range $0<\alpha<{4\over {N-2}}$, for every $\mu>0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0(x)=\mu|x|^{-{2\over\alpha}}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.