Non-uniqueness of positive solutions for supercritical semilinear heat equations without scale invariance

We establish nonuniqueness of solutions for Cauchy problems of semilinear heat equations with a wide class of nonlinearities. Specifically, we consider $$\{\begin{array}{ll}{\mathrm{\partial }}_{t}u-\mathrm{\Delta }u=f(u),& x\in {\mathbb{R}}^N,t>0,\\ u(x,0)=u_0(x),& x\in {\mathbb{R}}^N,\end{array}$$ 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 $u^{\ast }$. We prove that if $u_0=u^{\ast }$, then the problem has at least two positive solutions, namely $u^{\ast }$ and $u(t)$ which satisfies $u(t)\in L_{loc}^{\mathrm{\infty }}(0,t_0;L^{\mathrm{\infty }}({\mathbb{R}}^N))$ for some $t_0>0$ and $$u(t)\to u^{\ast }\text{in }{L}_{ul}^{\gamma }({\mathbb{R}}^N)\text{as }t\to 0^{+}$$ for $1\le \gamma <N(p_f-1)\mathrm{/}2$, where $p_f:={\lim }_{u\to \mathrm{\infty }}u{f}^{\mathrm{\prime }}(u)\mathrm{/}f(u)$ is a growth rate of $f$. Hence, nonuniqueness problem can be reduced to the existence problem of a positive radial singular stationary solution. The method of construction of $u(t)$ is based on the monotonicity argument. Transformations of forward self-similar solutions for $f(u)=u^p$ and $e^u$ play a crucial role.