AI 中文总结
该研究针对对数拉普拉斯驱动的半线性热方程，建立了适配非积分型对数热核的适定性理论，明确了寿命二分性、阈值现象及幂次非线性项的尖锐爆破速率。
AI 中文摘要
我们研究如下半线性热方程的非负温和解：∂_t u + (-Δ)^ln u = f(u)，定义域为(0,T)×ℝ^N，初值为u(0,·)=μ u_0，其中μ>0。与经典和分数阶热核不同，正对数热核仅在0<t<N/2时存在，对应的线性演化在终端时间可能出现奇异性，其最大寿命和终端增长均依赖于u_0的空间衰减性。f在零点附近的行为决定局部可解性：若∫_{0^+} dσ/f(σ) < ∞，则不存在定义在任意正时间区间上的非负有限解。在条件(𝒰_α)和(ℱ)下，我们发展了适配非积分型对数热核的适定性理论，还证明了两个互补的寿命判据：f整体最多线性增长时可获得完整线性寿命，而∫_{s_*}^∞ dσ/f(σ) < ∞意味着当μ→∞时最大存在时间趋于0。随后我们建立了慢衰减、快衰减和临界尾态下的寿命二分性：非临界态下，加权Osgood尾条件会引发过早爆破；在条件(ℱ_∞)下该条件不成立时则产生振幅阈值；临界尾态数据下，类似的二分性基于平方根加权Osgood条件成立，分界幂次为3/2。最后，我们推导了幂次非线性项的终端时间估计和尖锐爆破速率。
英文摘要
We investigate nonnegative mild solutions of $\partial_t u+(-Δ)^{\ln}u=f(u)$ in $(0,T)\times\mathbb R^N$, with initial datum $u(0,\cdot)=μu_0$, $μ>0$. Unlike the classical and fractional heat semigroups, the positive logarithmic heat kernel exists only for $0<t<N/2$, and the corresponding linear evolution may become singular at its terminal time, with lifespan and growth depending on the spatial decay of $u_0$. The behavior of $f$ near zero determines local solvability: if $\int_{0^+} dσ/f(σ)<\infty$, then no finite nonnegative solution exists on any positive time interval. Under suitable assumptions on $u_0$ and $f$, we establish well-posedness for the nonintegrable logarithmic heat kernel. If $f$ has at most global linear growth, the nonlinear solution attains the full linear lifespan, while the Osgood condition at infinity implies that the maximal existence time tends to zero as $μ\to\infty$. We further distinguish slow-decay, fast-decay, and critical-tail initial data. In the noncritical regimes, a weighted Osgood tail condition yields blow-up strictly before the linear terminal time; if it fails, an amplitude threshold occurs under additional assumptions on $f$. In the critical regime, the dividing power is $3/2$: a square-root weighted Osgood condition yields premature blow-up, while its failure again leads to an amplitude threshold. Finally, we obtain terminal-time blow-up estimates and sharp rates for power nonlinearities.
