| Abstract: |
| We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities.
Specifically, we consider
\[
\begin{cases}
\partial_tu-\Delta u=f(u), & x\in {\bf R}^N,\ t>0,\
u(x,0)=u_0(x), & x\in {\bf R}^N,
\end{cases}
\]
where $N>2$.
We assume that the growth rate of $f$ is less than the Joseph-Lundgren exponent for $N>10$ and it satisfies certain assumptions guaranteeing a positive radial singular stationary solution $u^*$.
We prove that if $u_0=u^*$, then the problem has at least two positive solutions, namely $u^*$ and $u(t)$ which satisfies $u(t)\in L_{loc}^{\infty}(0,t_0;L^{\infty}({\bf R}^N))$ for some $t_0>0$ and
$$
u(t)\to u^*\quad\text{in}\ L^{\gamma}_{ul}({\bf R}^N)\quad\text{as}\ t\to 0^+
$$
for $1\le \gamma |
|