On the existence and nonexistence of global solutions of the semilinear heat equation

We consider the semilinear heat equation $$u_t-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^{p-1}u,(t,x)\in {\mathbb{R}}^{+}\times {\mathbb{R}}^n\mathrm{.}$$ The well-known difficulty with this problem is that the potential well method cannot be applied directly, due to the scaling invariance which leads to a potential well of zero depth. We employ the forward similarity transform to convert the equation into a new parabolic equation, so that we can apply the potential well method in weighted Sobolev spaces. As a result, we obtain a new criterion that establishes whether solutions to the heat equation blow up in finite time or exist globally. This work extends the partial results of Ikehata et al. ($\mathit{\text{Ann.}}\mathit{\text{Inst.}}\mathit{\text{H.}}\mathit{\text{Poincar}}\acute{\mathit{\text{e}}}\mathit{\text{Anal.}}\mathit{\text{Non}}\mathit{\text{Lin}}\acute{\mathit{\text{e}}}\mathit{\text{aire}}$, $\mathbf{\text{27}}$ (2010) 877-900) from critical Sobolev exponent to the case $p_F<p<p_S$, where $p_F=1+2\mathrm{/}n$ is the Fujita exponent and $p_S=(n+2)\mathrm{/}(n-2)$ (for $n\ge 3$) is the critical Sobolev exponent.