论文标题
在非线性边界条件下，对热方程的溶液的寿命迅速估计
Sharp estimate of the life span of solutions to the heat equation with a nonlinear boundary condition
论文作者
论文摘要
考虑具有非线性边界条件的热方程$ \ partial_t u =ΔU，\ quad x \ in {\ bf r}^n _+，\，\，\，\，\，\，\，\ qquad \ qquad \ ppartial_νu= u^p， r}^n _+，\，\，\，\，t> 0，\ qquad u（x，x，0）=κψ（x），\ quad x \ in d：= \ overline {{\ bf r}^n_+} $ {\ bf r}^n_+：= \ {y \ in {\ bf r}^n：y_n> 0 \} $。让我们用$ t（κψ）$解决这个问题的寿命。我们研究了原始$ψ$的奇异性与足够大的$κ> 0 $的$ψ$与$ t（κψ）$与太空无穷大的$ψ$的行为与$ t（κψ）$之间的关系，对于足够小的$κ> 0 $。此外，我们给出$ t（κψ）$的最佳估计，为$κ\ to \ infty $或$κ\ to+0 $。
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 $κ\to+0$.
