Perturbations of self-similar solutions

We consider the nonlinear heat equation $u_t=\mathrm{\Delta }u+\mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u$ with $\alpha >0$, either on ${\mathbb{R}}^N$, $N\ge 1$, or on a bounded domain with Dirichlet boundary conditions. We prove that in the Sobolev subcritical case $(N-2)\alpha <4$, for every $\mu \in \mathbb{R}$, if the initial value $u_0$ satisfies $u_0(x)=\mu \mathrm{\mid }x-x_0{\mathrm{\mid }}^{-\frac{2}{\alpha }}$ in a neighborhood of some $x_0\in \mathrm{\Omega }$ and is bounded outside that neighborhood, then there exist infinitely many solutions of the heat equation with the initial condition $u(0)=u_0$. The proof uses a fixed-point argument to construct perturbations of self-similar solutions with initial value $\mu |x-x_0|^{-\frac {2} {\alpha }}$on${\mathbb R}^N $. Moreover, if $\mu \ge {\mu }_0$ for a certain ${\mu }_0(N,\alpha )\ge 0$, and $u_{0}I\ge 0$, then there is no nonnegative local solution of the heat equation with the initial condition $u(0)=u_0$, but there are infinitely many sign-changing solutions.