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 $$ {\left\{ \begin{array}{ll} \partial _tu-\Delta u=f(u), & x\in \mathbb {R}^N,\ t>0,\ u(x,0)=u_0(x), & x\in \mathbb {R}^N, \end{array}\right. } $$ 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 the existence of 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 $$u(t)\in L_{loc}^{\infty }(0,t_0;L^{\infty }(\mathbb {R}^N))$$ for some $$t_0>0$$ and $$ u(t)\rightarrow u^*\quad \text {in}\ L^{\gamma }_{ul}(\mathbb {R}^N)\quad \text {as}\ t\rightarrow 0^+ $$ for $$1\le \gamma <N(p_f-1)/2$$ , where $$p_f:=\lim _{u\rightarrow \infty }uf'(u)/f(u)$$ 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.