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arXiv 2002.10806math.AP
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具有非线性边界条件的heat equation方程解的寿命的精确估计

Sharp estimate of the life span of solutions to the heat equation with a nonlinear boundary condition

Kotaro Hisa

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摘要：

考虑具有非线性边界条件$$ \partial_t u=Δu,\quad x\in{\bf R}^N_+,\,\,\,t>0,\qquad \partial_νu=u^p, \quad x\in\partial{\bf R}^N_+,\,\,\,t>0,\qquad u(x,0)=κψ(x),\quad x\in D:=\overline{{\bf R}^N_+}, $$的热方程，其中$N\ge 1$，$p>1$，$κ>0$和$ψ$是${\bf R}^N_+ :=\{y\in{\bf R}^N:y_N>0 \}$中的非负可测函数.让我们用$T(κψ)$来表示这个问题的解决方案的寿命。我们研究了$ψ$在原点处的奇异性与足够大的$κ>0$的$T(κψ)$之间的关系，以及$ψ$在空间无穷远处的行为与足够小的$κ>0$的$T(κψ)$之间的关系。此外，我们还给出了$T(κψ)$的一个最优估计：$κ\to\infty$或$κ\to+0$ .

英文摘要：

Consider the heat equation with a nonlinear boundary condition $$ \partial_t u=Δu,\quad x\in{\bf R}^N_+,\,\,\,t>0,\qquad \partial_νu=u^p, \quad x\in\partial{\bf R}^N_+,\,\,\,t>0,\qquad u(x,0)=κψ(x),\quad x\in D:=\overline{{\bf R}^N_+}, $$ where $N\ge 1$, $p>1$, $κ>0$ and $ψ$ is a nonnegative measurable function in ${\bf R}^N_+ :=\{y\in{\bf R}^N:y_N>0 \}$. Let us denote by $T(κψ)$ the life span of solutions to this problem. We investigate the relationship between the singularity of $ψ$ at the origin and $T(κψ)$ for sufficiently large $κ>0$ and the relationship between the behavior of $ψ$ at the space infinity and $T(κψ)$ for sufficiently small $κ>0$. Moreover, we give an optimal estimate to $T(κψ)$, as $κ\to\infty$ or $κ\to+0$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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