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

Abstract

Abstract:

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

$ \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. $

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

CLC Number:

O175.29

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

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