具有强非均匀吸收的多孔介质方程的永恒解。第二部分：死核剖面
Eternal solutions to a porous medium equation with strong nonhomogeneous absorption. Part II: Dead-core profiles
AI总结：
针对带强非均匀吸收的多孔介质方程，本文构造并证明指数自相似永恒解在唯一指数 β_* 下存在带死核的紧支集剖面族，并阐明其界面行为及有限时间熄灭的最优界限。
摘要：
本文证明了如下具有强吸收的多孔介质方程存在一类指数自相似形式的特定永恒解族：方程为 ∂_t u−Δu^m+|x|^σu^q=0，定义于 (0,∞)×R^N，其中 m>1，q∈(0,1)，且 σ=2(1−q)/(m−1)。通过寻找形如 u(t,x)=e^{−αt}f(|x|e^{βt})、α=2β/(m−1) 的解，结果表明：当 m+q>2 时，存在唯一指数 β_*∈(0,∞)，使方程具有一个呈现死核的紧支集剖面单参数族。文中还确定了解在界面处的精确行为。此外，这些解揭示了以往研究所讨论的 Cauchy 问题真正非负解有限时间熄灭性质的最优限制。
英文摘要：
Existence of a specific family of \emph{eternal solutions} in exponential self-similar form is proved for the following porous medium equation with strong absorption $$\partial_t u-Δu^m+|x|^σu^q = 0 \;\;\text{ in }\;\; (0,\infty)\times\mathbb{R}^N,$$ with $m>1$, $q\in(0,1)$ and $σ=2(1-q)/(m-1)$. Looking for solutions of the form $$ u(t,x)=e^{-αt}f(|x|e^{βt}), \qquad α=\frac{2}{m-1}β,$$ it is shown that, for $m+q>2$, there exists a unique exponent $β_*\in(0,\infty)$ for which there exists a one-parameter family of compactly supported profiles presenting a \emph{dead core}. The precise behavior of the solutions at their interface is also determined. Moreover, these solutions show the optimal limitations for the finite time extinction property of genuine non-negative solutions to the Cauchy problem, studied in previous works.
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