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  

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

作　　者：沈旭辉 丁俊堂[2] Shen Xuhui;Ding Juntang(School of Applied Mathematics,Shanxi University of Finance and Economics,Taiyuan 030006;School of Mathematical Sciences,Shanxi University,Taiyuan 030006)

机构地区：[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

出　　处：《数学物理学报（A辑）》2023年第5期1417-1426,共10页Acta Mathematica Scientia

基　　金：国家自然科学基金(61473180);山西省青年科技研究基金(20210302124533)。

摘　　要：该文主要分析下列多孔介质方程组解的爆破现象{u_(t)=Δu^(l)+f(u,v,|∇u|^(2),t),v_(t)=Δv^(m)+g(u,v,|∇v|^(2),t),x∈Ω,t∈(0,t^(∗)),∂u/∂ν=p(u),∂v/∂ν=q(v),x∈∂Ω,t∈(0,t^(∗)),u(x,0)=u_(0)(x),v(x,0)=v_(0)(x),x∈¯Ω,其中l,m>1,Ω⊂R^(N)(N≥2)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.In this paper,we consider the blow-up of solutions to the following porous medium systems:{u_(t)=Δu^(l)+f(u,v,|∇u|^(2),t),v_(t)=Δv^(m)+g(u,v,|∇v|^(2),t),x∈Ω,t∈(0,t^(∗)),∂u/∂ν=p(u),∂v/∂ν=q(v),x∈∂Ω,t∈(0,t^(∗)),u(x,0)=u_(0)(x),v(x,0)=v_(0)(x),x∈¯Ω,where l,m>1,Ω⊂RN(N≥2)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.

关 键 词：多孔介质方程组 爆破 非线性边界条件 

分 类 号：O175.29[理学—数学]