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Qualitative theory and asymptotic analysis of nonlinear partial differential equations

非线性偏微分方程的定性理论与渐近分析

基本信息

批准号：
13440028
负责人：
MATANO Hiroshi
金额：
$ 10.3万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (B)
财政年份：
2001
资助国家：
日本
起止时间：
2001 至 2004
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-13440028/
关键词：
nonlinear partial differential equation qualitative theory asymptotic analysis blow-up of solutions singularity singular limit traveling wave inverse problem 方程式の縮約自由境界

项目摘要

We have considered the behavior of solutions after the blow-up time for nonlinear heat equations with a power nonlinearity and those with an exponential nonlinearity. It is shown that singularities that apper at the blow-up time disappear and the solutions become smooth immediately (Matano, SIAM J.Math.Anal., in press). We have also studied the blow-up rate for nonlinear heat equations with a power nonlinearity and proved that the blow-up is always type 2 so far as the power is in the intermediate supercritical range (Matano, Comm.Pure Appl.Math., 2004).Yamamoto has studied an inverse problem of determining two unknown convection terms in a two-dimensional elliptic equation, and proved that those terms can be determined by the so-called Dirichlet-Neumann map (Inverse Problems, 2004).Weiss has considered a singular limit problem for parabolic equations that are applicable to the double obstacle problem. Using a new monotonicity formula, he has succeeded in exstimating the Hausdorff dimension of the free boundary (Calc.Var.PDE, 2003).Ei has studied a reaction-diffusion system on a two-dimensional cylinder and analysed the behavior of solutions having a pulse-like profile. He derived an equation governing the motion of slow pulses and proved that traveling pulses are mutually repelling (DCDS, Ser.A, in press).Taniguchi has considered the so-called "singular limit eigenvalue problem method" (SLEP method), which is a powerful tool in analyzing the stability of stationary solution of the singular limit problem. He has generalized this method so that it applies to problems in unbounded domains. Using this result, he has proved the stability of planar traveling waves in a bistable reaction-diffusion system (DCDS, Ser.B, 2003).

研究了一类具有幂非线性项和指数非线性项的非线性热方程解在爆破时刻后的行为。结果表明，在爆破时出现的奇异性消失，解立即变得光滑（Matano，SIAM J.Math.Anal.，印刷中）。我们还研究了具有幂非线性项的非线性热方程的爆破速率，证明了只要幂在中超临界范围内，爆破总是2型的（Matano，Comm.Pure Appl.Math.，2004). Yamamoto研究了确定二维椭圆型方程中两个未知对流项的反问题，并证明了这些项可以通过所谓的Dirichlet-Neumann映射确定（Inverse Problems，2004）。韦斯考虑了适用于双障碍问题的抛物型方程的奇异极限问题。利用一个新的单调性公式，他成功地计算了自由边界的Hausdorff维数（Calc.Var.PDE，2003）.Ei研究了二维圆柱上的反应扩散系统，并分析了具有脉冲状轮廓的解的行为.他导出了慢脉冲运动的方程，并证明了行波脉冲是相互排斥的（DCDS，Ser.A，in press）。谷口考虑了所谓的“奇异极限特征值问题方法”（SLEP方法），这是分析奇异极限问题稳定解的有力工具。他推广了这种方法，使其适用于无界域的问题。利用这个结果，他证明了平面行波在一个非线性反应扩散系统中的稳定性（DCDS，Ser.B，2003）。