An Exterior Parabolic Differential Inequality Under Semilinear Dynamical Boundary Conditions,Bulletin of the Malaysian Mathematical Sciences Society

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An Exterior Parabolic Differential Inequality Under Semilinear Dynamical Boundary Conditions
Bulletin of the Malaysian Mathematical Sciences Society ( IF 1.0 ) Pub Date : 2020-07-15 , DOI: 10.1007/s40840-020-00973-0
Awatif Alqahtani , Mohamed Jleli , Mokhtar Kirane , Bessem Samet

We study the existence and nonexistence of global weak solutions to the semilinear parabolic differential inequality

$$\begin{aligned} \partial _t u-\Delta u \ge |u|^p,\quad (t,x)\in (0,\infty )\times B^c, \end{aligned}$$

where $p>1$, B is the closed unit ball in ${\mathbb {R}}^N$ ($N\ge 2$) and $B^c$ is its complement, under the semilinear dynamical boundary conditions

$$\begin{aligned} \partial _t u+u \ge |u|^q +w(x), \quad (t,x)\in (0,\infty )\times \partial B \end{aligned}$$

$$\begin{aligned} \partial _t u+\partial _\nu u +\alpha u \ge |u|^q +w(x), \quad (t,x)\in (0,\infty )\times \partial B, \end{aligned}$$

where $q>1$, $\alpha \ge 0$, $\partial _\nu :=\frac{\partial }{\partial \nu ^+}$, $\nu ^+$ is the outward unit normal (relative to $B^c$) on $\partial B$ and $w\in L^1(\partial B)$, $\int _{\partial B} w(x)\,\hbox {d}S_x\ge 0$. The cases $\int _{\partial B} w(x)\,\hbox {d}S_x= 0$ and $\int _{\partial B} w(x)\,\hbox {d}S_x>0$ are discussed separately.

中文翻译：

半线性动力边界条件下的外抛物型不等式

我们研究半线性抛物型微分不等式的全局弱解的存在和不存在

$$ \ begin {aligned} \ partial _t u- \ Delta u \ ge | u | ^ p，\ quad（t，x）\ in（0，\ infty）\ times B ^ c，\ end {aligned} $ $

其中\（p> 1 \），B是\（{\ mathbb {R}} ^ N \）（\（N \ ge 2 \））中的闭合单位球，\（B ^ c \）是其补码在半线性动力边界条件下

$$ \ begin {aligned} \ partial _t u + u \ ge | u | ^ q + w（x），\ quad（t，x）\ in（0，\ infty）\ times \ partial B \ end {aligned } $$

要么

$$ \ begin {aligned} \ partial _t u + \ partial _ \ nu u + \ alpha u \ ge | u | ^ q + w（x），\ quad（t，x）\ in（0，\ infty）\ \ partial B，\ end {aligned} $$

其中\（q> 1 \），\（\ alpha \ ge 0 \），\（\ partial _ \ nu：= \ frac {\ partial} {\ partial \ nu ^ +} \），\（\ nu ^ + \）是向外单位法线（相对于\（B ^ C \））上\（\局部乙\）和\（W \在L ^ 1（\部分B）\），\（\ INT _ { \ partial B} w（x）\，\ hbox {d} S_x \ ge 0 \）。的情况下\（\ INT _ {\部分B} W（x）的\，\ hbox中{d} S_x = 0 \）和\（\ INT _ {\部分B} W（x）的\，\ hbox中{d} S_x> 0 \）分开讨论。

更新日期：2020-07-15

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