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

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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. 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 Juntang2,*()   

  1. 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:

该文主要分析下列多孔介质方程组解的爆破现象

{ut=Δul+f(u,v,|u|2,t),vt=Δvm+g(u,v,|v|2,t),xΩ, t(0,t),uν=p(u), vν=q(v),xΩ, t(0,t),u(x,0)=u0(x), v(x,0)=v0(x),x¯Ω,

其中 l,m>1, ΩRN (N2) 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:

{ut=Δul+f(u,v,|u|2,t),vt=Δvm+g(u,v,|v|2,t),xΩ, t(0,t),uν=p(u), vν=q(v),xΩ, t(0,t),u(x,0)=u0(x), v(x,0)=v0(x),x¯Ω,

where l,m>1, ΩRN (N2) is a bounded domain with smooth boundary Ω. 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