发表机构
Institute of Mathematics and Mathematical Modeling; SDU University（数学与数学建模研究所; SDU大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对零质量变号初值半线性热方程，本文识别出新的过渡指数并建立尖锐寿命估计，揭示了与经典Lee-Ni定律不同的寿命现象，扩展了Fujita临界指数理论。
AI 中文摘要
我们考虑Cauchy问题 $$u_t-\Delta u=|u|^p,\qquad (t,x)\in(0,T)\times\R^n, \quad u(0,x)=\eps u_0(x),\qquad x\in\R^n,$$ 其中初值具有小振幅、变号且总质量为零。我们假设 $$u_0\in L^1(\R^n)\cap L^\infty(\R^n),\quad |x|u_0\in L^1(\R^n),\quad \int_{\R^n}u_0(x)\\,dx=0,$$ 并且对于尖锐次临界上界，还假设 $\int_{\R^n}x\\,u_0(x)\\,dx\neq0.$ 设 $T_\eps$ 为最大寿命。我们在Fujita范围 $1<p\le p_F:=1+2/n$ 内识别出一个新的过渡指数 $p_m=1+\frac1{n+1}$，并建立了尖锐的寿命估计 $$T_\eps\asymp \begin{cases} \eps^{-\frac{2(p-1)}{2-(n+1)(p-1)}}, &1<p<p_m,\\\\[2mm] \eps^{-\frac2{n+1}} \bigl(\log\frac1\eps\bigr)^{-\frac2{n+2}}, &p=p_m,\\\\[2mm] \eps^{-p\left(\frac{1}{p-1}-\frac{n}{2}\right)^{-1}}, &p_m<p<p_F,\\\\[2mm] \exp\\!\left(\eps^{-p(p-1)}\right), & p=p_F. \end{cases}$$ 这些估计揭示了一种不同于经典Lee--Ni定律（针对正质量初值）的寿命现象。在零质量情形下，主导的线性贡献是偶极型的，而非线性源随后产生大小为 \\(O(\eps^p)\\) 的正质量。它们的竞争产生了额外的阈值 \\(p_m\\)、在 \\(p=p_m\\) 处的对数修正，以及在 \\(p=p_F\\) 处显著更长的临界寿命，其指数为 \\(p_F(p_F-1)\\) 而非经典的Lee--Ni指数 \\(p_F-1\\)。上界估计通过后向高斯和临界尺度ODE测试函数论证获得，而下界估计则通过统一的 \\(L^1\\)--\\(L^\infty\\) 自举方法得到，该方法保持了线性流的零质量抵消并控制了非线性源产生的质量。因此，尽管 \\(p_F\\) 仍然是Fujita临界指数，零初始质量在临界指数以下及临界指数处创造了新的定量寿命机制。
英文摘要
We consider the Cauchy problem $$u_t-Δu=|u|^p,\qquad (t,x)\in(0,T)\times\R^n, \quad u(0,x)=\eps u_0(x),\qquad x\in\R^n,$$ 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.
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