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

RESUMO

We consider the Cauchy problem $$u_t-\Delta u=|u|^p,\qquad (t,x)\in(0,T)\times\R^n, \quad u(0,x)=\eps u_0(x),\qquad x\in\R^n,$$ with small sign-changing initial data having zero total mass. We assume $$u_0\in L^1(\R^n)\cap L^\infty(\R^n),\quad |x|u_0\in L^1(\R^n),\quad \int_{\R^n}u_0(x)\,dx=0,$$ and, for the sharp subcritical upper bounds, that $\int_{\R^n}x\,u_0(x)\,dx\neq0.$ Let $T_\eps$ denote the maximal lifespan. We identify a new transition exponent $p_m=1+\frac1{n+1}$ inside the Fujita range $1<p\le p_F:=1+2/n$, and establish the sharp lifespan estimates $$T_\eps\asymp \begin{cases} \eps^{-\frac{2(p-1)}{2-(n+1)(p-1)}}, &1<p<p_m,\\[2mm] \eps^{-\frac2{n+1}} \bigl(\log\frac1\eps\bigr)^{-\frac2{n+2}}, &p=p_m,\\[2mm] \eps^{-p\left(\frac{1}{p-1}-\frac{n}{2}\right)^{-1}}, &p_m<p<p_F,\\[2mm] \exp\!\left(\eps^{-p(p-1)}\right), & p=p_F. \end{cases}$$ 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 \(O(\eps^p)\). 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 $L^1$--$L^\infty$ 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.