Existence and Blowup of Positive Solutions to a Class of Multilateral Flow Equations
-
摘要: 该文研究了一类具有非局部Neumann边界条件和非线性吸收项的多方渗流方程解的全局存在性和爆破情况.首先针对所研究方程定义了其上下解,并建立和证明了比较原理;然后通过构造函数以及利用微分不等式、特征值特征函数、常微分方程的解和椭圆第二边值的解等方法对方程进行了研究,得到了对于不同取值范围的参数、权函数和初始值时,方程非负解的全局存在性和在有限时间内爆破的充分条件.
-
关键词:
- Neumann边界条件 /
- 多方渗流方程 /
- 全局存在性 /
- 爆破
Abstract: The global existence and blowup of the solutions to a class of multilateral filtration equations with non-local Neumann boundary conditions and nonlinear absorption terms were studied. First, the super- and sub-solutions were defined for the studied equations and the comparison principle was established. Then, the equation was investigated with constructed functions, differential inequalities, eigenfunctions, ordinary differential equation and elliptic second boundary value solutions. The global existence of non-negative solutions to the equations and the conditions for blowup in a finite time for the parameters, weight functions and initial values in different value ranges were obtained.-
Key words:
- Neumann boundary condition /
- multilateral filtration equation /
- global existence /
- blowup
-
[2]朱位秋. 几类非线性系统对白噪声参激与/或外激平稳响应的精确解[J]. 应用数学和力学, 1990,11(2): 155-164.(ZHU Weiqiu. Exact solutions for stationary responses of several classes of nonlinear systems to parametric and/or external white noise excitations[J].Applied Mathematics and Mechanics,1990,11(2): 155-164.(in Chinese)) POINSOT T J, LELE S K. Boundary conditions for direct simulations of compressible viscous flows[J].Journal of Computational Physics,1992,101(1): 104-129. [3]GLADKOV A, KIM K I. Blow-up of solutions for semilinear heat equation with nonlinear nonlocal boundary condition[J].Journal of Mathematical Analysis and Applications,2008,338(1): 264-273. [4]WANG Y, MU C, XIANG Z. Blowup of solutions to a porous medium equation with nonlocal boundary condition[J].Applied Mathematics & Computation,2007,192(2): 579-585. [5]YE Z, XU X J. Global existence and blow-up for a porous medium system with nonlocal boundary conditions and nonlocal sources[J].Nonlinear Analysis,2013, 82: 115-126. [6]LI Y H, MI Y S, MU C L. Properties of positive solutions for a nonlocal nonlinear diffusion equation with nonlocal nonlinear boundary condition[J].Acta Mathematica Scientia,2014,34(3): 748-758. [7]张正策, 王彪. 含有非线性梯度项的退化抛物方程解的爆破率估计[J]. 应用数学和力学, 2010,31(6): 756-764.(ZHANG Zhengce, WANG Biao. Blow-up rate estimate for degenerate parabolic equation with nonlinear gradient term[J].Applied Mathematics and Mechanics,2010, 31(6): 756-764.(in Chinese)) [8]GLADKOV A, KAVITOTA T. Blow-up problem for semilinear heat equation with nonlinear nonlocal boundary condition[J].Applicable Analysis,2016, 95(9): 1974-1988. [9]ZHOU S, YANG Z D. Blow-up of solutions for a reaction-diffusion equation with nonlinear nonlocal boundary condition[J].British Journal of Mathematics & Computer Science,2016,16: 1-9. [10]WANG J, YANG H. Properties of solutions for a reaction-diffusion equation with nonlinear absorption and nonlinear nonlocal Neumann boundary condition[J].Boundary Value Problems,2018,2018. DOI: 10.1186/s13661-018-1069-9. [11]LIU B, WU G, SUN X, et al. Blow-up estimate in a reaction-diffusion equation with nonlinear nonlocal flux and source[J].Computers and Mathematics With Applications,2019,78(6): 1862-1877. [12]GLADKOVA A, GUEDDA M. Global existence of solutions of a semilinear heat equation with nonlinear memory condition[J].Applicable Analysis,2020,99(16): 2823-2832. [13]LIU B, LIN H, LI F, et al. Blow-up analyses in reaction-diffusion equations with nonlinear nonlocal boundary flux[J].Zeitschrift für Angewandte Mathematik und Physik,2019, 70(4): 106-133. [14]王明新. 非线性抛物型方程[M]. 北京: 科学出版社, 1993.(WANG Mingxin.Nonlinear Parabolic Equation[M]. Beijing: Science Press, 1993.(in Chinese)) [15]ANDERSON J R. Local existence and uniqueness of solutions of degenerate parabolic equations[J].Communications in Partial Differential Equations,1991,16(1): 105-143. [16]GLADKOV A. Initial boundary value problem for a semilinear parabolic equation with absorption and nonlinear nonlocal boundary condition[J].Lithuanian Mathematical Journal,2017,57(4): 468-478.
点击查看大图
计量
- 文章访问数: 470
- HTML全文浏览量: 104
- PDF下载量: 54
- 被引次数: 0