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.

参考文献 24

[1]	Protter M H, Weinberger H F. Maximum Principles in Differential Equations. Englewood Cliffs: Prentice-Hall, 1967
[2]	Friedman A. Partial Differential Equation of Parabolic Type. Englewood Cliffs: Prentice-Hall, 1964
[3]	Akagi G, Melchionna S. Porous medium equation with a blow-up nonlinearity and a non-decreasing constraint. Nonlinear Differ Equ Appl, 2019, 26(2): Article 10
[4]	Anderson J R, Deng K. Global solvability for the porous medium equation with boundary flux governed by nonlinear memory. J Math Anal Appl, 2015, 423(2): 1183-1202 doi: 10.1016/j.jmaa.2014.10.041
[5]	Andreu F, Mazón J M, Toledo J, Rossi J D. Porous medium equation with absorption and a nonlinear boundary condition. Nonlinear Anal TMA, 2002, 49(4): 541-563 doi: 10.1016/S0362-546X(01)00122-5
[6]	Ding J T, Shen X H. Blow-up time estimates in porous medium equations with nonlinear boundary conditions. Z Angew Math Phys, 2018, 69(4): Article 99
[7]	Iagar R G, Sánchez A, Blow up profiles for a quasilinear reaction-diffusion equation with weighted reaction. J Differential Equations, 2021, 272: 560-605 doi: 10.1016/j.jde.2020.10.006
[8]	Li F C, Xie C H. Global existence and blow-up for a nonlinear porous medium equation. Appl Math Lett, 2003, 16(2): 185-192 doi: 10.1016/S0893-9659(03)80030-7
[9]	Liang Z L. Blow up rate for a porous medium equation with power nonlinearity. Nonlinear Anal TMA, 2010, 73(11): 3507-3512 doi: 10.1016/j.na.2010.06.078
[10]	Schaefer P W. Blow-up phenomena in some porous medium problems. Dynam Systems Appl, 2009, 18(1): 103-109
[11]	Tian H M, Zhang L L. Blow-up solution of a porous medium equation with nonlocal boundary conditions. Complexity, 2020, 2020: Article ID 9037287
[12]	Wang Z Y, Yin J X. Note on blow-up of solutions for a porous medium equation with convection and boundary flux. Colloq Math, 2012, 128(2): 223-228 doi: 10.4064/cm128-2-7
[13]	Wu X L, Gao W J. Blow-up of the solution for a class of porous medium equation with positive initial energy. Acta Math Sci, 2013, 33B(4): 1024-1030
[14]	Zhou J. A multi-dimension blow-up problem to a porous medium diffusion equation with special medium void. Appl Math Lett, 2014, 30: 6-11 doi: 10.1016/j.aml.2013.12.003
[15]	Du L L. Blow-up for a degenerate reaction-diffusion system with nonlinear localized sources. J Math Anal Appl, 2006, 324(1): 304-320 doi: 10.1016/j.jmaa.2005.11.052
[16]	Lei P D, Zheng S N. Global and nonglobal weak solutions to a degenerate parabolic system. J Math Anal Appl, 2006, 324(1): 177-198 doi: 10.1016/j.jmaa.2005.12.012
[17]	Li Y X, Gao W J, Han Y Z. Boundedness of global solutions for a porous medium system with moving localized sources. Nonlinear Anal TMA, 2010, 72(6): 3080-3090 doi: 10.1016/j.na.2009.11.047
[18]	Mu C L, Hu X G, Li Y H, Cui Z J. Blow-up and global existence for a coupled system of degenerate parabolic equations in a bounded domain. Acta Math Sci, 2007, 27B(1): 92-106
[19]	Mu C L, Su Y. Global existence and blow-up for a quasilinear degenerate parabolic system in a cylinder. Appl Math Lett, 2001, 14(6): 715-723 doi: 10.1016/S0893-9659(01)80032-X
[20]	Quir${\rm \acute{o}}$s J D, Rossi J D. Blow-up sets and Fujita type curves for a degenerate parabolic system with nonlinear boundary conditions. Indiana Univ Math J, 2001, 50(1): 629-654 doi: 10.1512/iumj.2001.50.1828
[21]	Shen X H, Ding J T. Blow-up phenomena in porous medium equation systems with nonlinear boundary conditions. Comput Math Appl, 2019, 77(12): 3250-3263 doi: 10.1016/j.camwa.2019.02.007
[22]	Wang Z J, Zhou Q A, Lou W Q. Critical exponents for porous medium systems coupled via nonlinear boundary flux. Nonlinear Anal TMA, 2009, 71(5/6): 2134-2140 doi: 10.1016/j.na.2009.01.047
[23]	Xia A Y, Fan M S, Li S. Blow-up and life span estimates for a class of nonlinear degenerate parabolic system with time-dependent coefficients. Acta Math Sci, 2017, 37B(4): 974-984
[24]	Payne L E, Philippin G A. Blow-up phenomena for a class of parabolic systems with time dependent coefficients. Appl Math, 2012, 3(4): 325-330