具有梯度源项和非线性边界条件的多孔介质方程组解的爆破

数学物理学报 ›› 2023, Vol. 43 ›› Issue (5): 1417-1426.

具有梯度源项和非线性边界条件的多孔介质方程组解的爆破

沈旭辉¹(),丁俊堂^2,^*()

¹山西财经大学应用数学学院太原 030006
²山西大学数学科学学院太原 030006

收稿日期:2022-07-14 修回日期:2023-03-23 出版日期:2023-10-26 发布日期:2023-08-09
通讯作者: 丁俊堂 E-mail:xhuishen@sxufe.edu.cn;djuntang@sxu.edu.cn
作者简介:沈旭辉，Email: xhuishen@sxufe.edu.cn
基金资助:
国家自然科学基金(61473180);山西省青年科技研究基金(20210302124533)

Blow-Up Conditions of Porous Medium Systems with Gradient Source Terms and Nonlinear Boundary Conditions

Shen Xuhui¹(),Ding Juntang^2,^*()

¹School of Applied Mathematics, Shanxi University of Finance and Economics, Taiyuan 030006
²School of Mathematical Sciences, Shanxi University, Taiyuan 030006

Received:2022-07-14 Revised:2023-03-23 Online:2023-10-26 Published:2023-08-09
Contact: Juntang Ding E-mail:xhuishen@sxufe.edu.cn;djuntang@sxu.edu.cn
Supported by:
NSFC(61473180);Youth Natural Science Foundation of Shanxi Province(20210302124533)

摘要/Abstract

摘要：

该文主要分析下列多孔介质方程组解的爆破现象

$\left\{ \begin{array}{ll} u_{t} =\Delta u^l+f(u,v,|\nabla u|^2,t), & \\\displaystyle v_{t} =\Delta v^m+g(u,v,|\nabla v|^2,t),&x\in\Omega, \ t\in(0,t^*), \\\displaystyle \frac{\partial u}{\partial\nu}=p(u), \ \frac{\partial v}{\partial\nu}=q(v), &x\in\partial\Omega, \ t\in(0,t^*), \\\displaystyle u(x,0)=u_{0}(x), \ v(x,0)=v_{0}(x), &x\in\overline{\Omega}, \end{array} \right.$

其中 $l, m>1, \ \Omega\subset\mathbb{R}^N \ (N\geq2)$ 为具有光滑边界的有界区域. 通过使用微分不等式技术和最大值原理, 给出方程组的解在有限时刻 $t^*$ 爆破的充分条件, 并分别导出了解的爆破时刻 $t^*$ 及爆破率的上估计.

关键词: 多孔介质方程组, 爆破, 非线性边界条件

Abstract:

In this paper, we consider the blow-up of solutions to the following porous medium systems:

where $l,m>1, \ \Omega\subset\mathbb{R}^N \ (N\geq2)$ is a bounded domain with smooth boundary $\partial\Omega$ . Using the differential inequality techniques and the maximum principles, we give a sufficient condition to ensure that the positive solution $(u,v)$ of the above problem is a blow-up solution that blows up at a certain finite time $t^*$ . An upper estimate of $t^*$ and an upper estimate of the blow-up rate of $(u,v)$ are also obtained.

Key words: Porous medium systems, Blow-up, Nonlinear boundary condition

中图分类号:

O175.29

沈旭辉,丁俊堂. 具有梯度源项和非线性边界条件的多孔介质方程组解的爆破[J]. 数学物理学报, 2023, 43(5): 1417-1426.

Shen Xuhui,Ding Juntang. Blow-Up Conditions of Porous Medium Systems with Gradient Source Terms and Nonlinear Boundary Conditions[J]. Acta mathematica scientia,Series A, 2023, 43(5): 1417-1426.

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