Generalized Solutions to Nonlinear Nonlocal Singularly Perturbed Parabolic Initial-Boundary Problems With Two Parameters
-
摘要: 研究了一类广义抛物型方程奇摄动问题.首先在一定的条件下, 提出了一类具有两参数的非线性非局部广义抛物型方程初始边值问题.其次证明了相应问题解的存在性.然后, 通过Fredholm积分方程得到了初始边值问题的外部解.再利用泛函分析理论和伸长变量及多重尺度法, 分别构造了初始边值问题广义解的边界层、初始层项,从而得到了问题的形式渐近展开式.最后利用不动点理论证明了对应的非线性非局部广义抛物型方程的奇异摄动初始边值问题的广义解的渐近展开式的一致有效性.Abstract: A class of generalized parabolic equation singular perturbation problems were considered. Firstly, under suitable conditions, a class of nonlinear nonlocal generalized parabolic equation initial-boundary value problems with two parameters were raised. Secondly, the existence of solutions to corresponding problems was proved. Next, from the Fredholm integral equation, the outer solutions to the initial-boundary value problems were found, and the boundary and initial layer terms were structured by means of the theory of functional analysis, the stretched variables and the multiscale methods, respectively. Then the formal asymptotic expansion of the problem was obtained. Finally, according to the fixed point theorem, the uniform validity of the asymptotic expansion of generalized solutions to the corresponding nonlinear nonlocal initial-boundary value problems was proved.
-
Key words:
- singular perturbation /
- asymptotic expansion /
- uniform validity
-
[1] Bartu L, Morosanu G. Singularly Perturbed Boundary-Value Problems[M]. Basel: Birkhauserm Verlag AG, 2007. [2] de Jager E M, JIANG Fu-ru. The Theory of Singular Perturbation [M]. Amsterdam: North-Holland Publishing Co, 1996. [3] Kellogg R B, Kopteva N. A singularly perturbed semi-linear reaction-diffusion problem in a polygonal domain[J].Journal of Differential Equations,2010,248 (1): 184-208. [4] TIAN Can-rong, ZHU Peng. Existence and asymptotic behavior of solutions for quasilinear parabolic systems[J].Acta Applicandae Mathematicae,2012,121(1): 157-173. [5] Samusenko P F. Asymptotic integration of degenerate singularly perturbed systems of parabolic partial differential equations[J].Journal of Mathematical Sciences,2013,189(5): 834-847. [6] Skrynnikov Y. Solving initial value problem by matching asymptotic expansions[J]. SIAM Journal on Applied Mathematics, 2012,72(1): 405-416. [7] Martinez S, Wolanski N. A singular perturbation problem for a quasi-linear operator satisfying the natural condition of Lieberman[J].SIAM J Math Anal,2009,41(1): 318-359. [8] MO Jia-qi. Singular perturbation for a class of nonlinear reaction diffusion systems[J]. Science in China(Ser A),1989,32(11): 1306-1315. [9] MO Jia-qi. Approximate solution of homotopic mapping to solitary wave for generalized nonlinear KdV system[J]. Chin Phys Lett, 2009,26(1): 010204. [10] MO Jia-qi, LIN Wan-tao. Generalized variation iteration solution of an atmosphere-ocean oscillator model for global climate[J]. Journal of Systems Science & Complexity, 2011,24(2): 271-276. [11] 冯依虎, 石兰芳, 汪维刚, 等. 一类广义非线性强阻尼扰动发展方程的行波解[J]. 应用数学和力学, 2015,36(3): 315-324.(FENG Yi-hu, SHI Lan-fang, WANG Wei-gang, et al. The traveling wave solution for a class of generalized nonlinear strong damping disturbed evolution equations[J]. Applied Mathematics and Mechanics,2015,36(3): 315-324.(in Chinese)) [12] 冯依虎, 石兰芳, 许永红, 等. 一类大气尘埃等离子体扩散模型研究[J]. 应用数学和力学, 2015,36(6): 639-650.(FENG Yi-hu, SHI Lan-fang, XU Yong-hong, et al. The study for a class of atomy plasma diffusion model in atmosphere[J]. Applied Mathematics and Mechanics,2015,36(6): 639-650.(in Chinese)) [13] 史娟荣, 朱敏, 莫嘉琪. 广义Schrdinger扰动耦合系统孤子解 [J]. 应用数学和力学, 2016,37(3): 319-330.(SHI Juan-rong, ZHU Min, MO Jia-qi. Solitary solutions to generalized Schrdinger disturbed coupled systems[J]. Applied Mathematics and Mechanics,2016,37(3): 319-330.(in Chinese))
点击查看大图
计量
- 文章访问数: 852
- HTML全文浏览量: 82
- PDF下载量: 456
- 被引次数: 0