New sharp lifespan estimates for a semilinear heat equation with zero-mass sign-changing initial data

We consider the Cauchy problem $$u_t-\mathrm{\Delta }u=\mathrm{\mid }u{\mathrm{\mid }}^p,(t,x)\in (0,T)\times {\mathbb{R}}^n,u(0,x)=\text{eps}{u}_0(x),x\in {\mathbb{R}}^n,$$ with small sign-changing initial data having zero total mass. We assume $$u_0\in L^1({\mathbb{R}}^n)\cap L^{\mathrm{\infty }}({\mathbb{R}}^n),\mathrm{\mid }x\mathrm{\mid }{u}_0\in L^1({\mathbb{R}}^n),{\int }_{{\mathbb{R}}^n}{u}_0(x)dx=0,$$ and, for the sharp subcritical upper bounds, that ${\int }_{{\mathbb{R}}^n}x{u}_0(x)dx\mathrm{\ne }0.$ Let $T_{\text{eps}}$ denote the maximal lifespan. We identify a new transition exponent $p_m=1+\frac{1}{n+1}$ inside the Fujita range $1<p\le p_F:=1+2\mathrm{/}n$, and establish the sharp lifespan estimates $$T_{\text{eps}}\asymp \{\begin{array}{ll}{\text{eps}}^{-\frac{2(p-1)}{2-(n+1)(p-1)}},& 1<p<p_m,\\ {\text{eps}}^{-\frac{2}{n+1}}(\log \frac{1}{\text{eps}}{)}^{-\frac{2}{n+2}},& p=p_m,\\ {\text{eps}}^{-p{(\frac{1}{p-1}-\frac{n}{2})}^{-1}},& p_m<p<p_F,\\ \exp \text{\hspace{0.17em}⁣}\left({\text{eps}}^{-p(p-1)}\right),& p=p_F\mathrm{.}\end{array}$$ These estimates reveal a lifespan phenomenon that is different from the classical Lee–Ni law for initial data with positive mass. In the zero-mass setting, the leading linear contribution is dipole-like, while the nonlinear source subsequently generates a positive mass of size $O({\text{eps}}^p)$. Their competition produces the additional threshold $p_m$, the logarithmic correction at $p=p_m$, and, at $p=p_F$, a substantially longer critical lifespan with exponent $p_F(p_F-1)$ instead of the classical Lee–Ni exponent $p_F-1$. The upper estimates are obtained by backward-Gaussian and critical scale-ODE test-function arguments, whereas the lower estimates follow from a unified $L^1$–$L^{\mathrm{\infty }}$ bootstrap preserving the zero-mass cancellation of the linear flow and controlling the mass generated by the nonlinear source. Thus, although $p_F$ remains the Fujita critical exponent, zero initial mass creates a new quantitative lifespan regime below and at the critical exponent.