On the behavior of solutions for nonlinear parabolic or elliptic equations

关于非线性抛物线或椭圆方程解的行为

• 批准号: 13640205
• 负责人: MIZOGUCHI Noriko
• 金额: $ 2.18万
• 依托单位: TOKYO GAKUGEI UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640205/
• 关键词: nonlinear; parabolic; blow-up; global solution; critical exponent; complementary space; stable domain

It seems that there have been no results on the blow-up of solutions for a semilinear heat equation with power nonlinearity under the Neumann boundary condition in a bounded domain since the well-known result by Giga and Kohn cannot work effectively, which is a quite difference from the Cauchy or the Dirichlet-Cauchy problem. I proved that the blow-up rate and behaviors of solutions are as same as that in the result by Giga and Kohn by a joint work with Kazuhiro Ishige of Nagoya University. Moreover, we showed that the bllow-up occurs only near the maximum points of the second eigenfunctions of the laplacian with the Nuemann boundary condition. This result is closely related to the hot spots conjecture for the heat equation.On the other hand, the structure of global solutions of the Cauchy problem for tie above equation changes suddenly near some exponent Yanagida obtained the global stability of stationary solutions. As a application, he showed the existence of unbounded global solutions which behave very complicatedly.Yamada investigated some fundamental properties about the complementary spaces on the linear contraction of de Branges and obtained a simple proof the problem on the extended interpolation in the case of the unit ball due to Takahashi of Nara University.Kubota studied about the stable domain of automorphism in a Banach space.Ito tried to apply the above studies to practical information education.

由于 Giga 和 Kohn 的著名结果无法有效地发挥作用，因此在有界域内的诺依曼边界条件下具有功率非线性的半线性热方程的解的爆炸似乎还没有结果，这是一个相当大的问题。与柯西或狄利克雷-柯西问题的区别。我与名古屋大学的 Kazuhiro Ishige 合作，证明了解决方案的爆炸率和行为与 Giga 和 Kohn 的结果相同。此外，我们表明，爆炸仅发生在具有努伊曼边界条件的拉普拉斯第二本征函数的极大点附近。这一结果与热方程的热点猜想密切相关。另一方面，上述方程的柯西问题全局解的结构在某个指数附近突然变化，柳田得到了平稳解的全局稳定性。作为一个应用，他展示了行为非常复杂的无界全局解的存在性。Yamada 研究了 de Branges 线性收缩上的互补空间的一些基本性质，并获得了单位情况下扩展插值问题的简单证明奈良大学高桥的球。久保田研究了巴拿赫空间中自同构的稳定域。伊藤试图将上述研究应用到实际的信息教育中。

项目成果

N.Mizoguchi, K.Ishige: "Blow-up behavior for semilinear heat equations with Neumann boundary conditions"J.Differential Integral Equations. (to appear).

N.Mizoguchi、K.Ishige：“具有诺伊曼边界条件的半线性热方程的爆炸行为”J.微分积分方程。

E.Yanagida, P.Polacik: "Existence of stable subharmonic solutions for reaction-diffusion equations"J.Differential Equations. 169. 255-280 (2001)

E.Yanagida、P.Polacik：“反应扩散方程稳定次谐波解的存在”J.微分方程。

Y.Kubota: "Stable domains of automorphisms of Banach spaces at fixed points"Bulletin of Tokyo Gakugei University, Sect.IV Mathematics and Natural Sciences. 53. (2001)

Y.Kubota：“定点巴纳赫空间自同构的稳定域”东京学艺大学通报，第四节数学和自然科学。

E. Yanagida, W. M. Ni and P. Polacik: "Monotonicity of stable solutions in shadow Systems"Trans. Amer. Math. Soc.. 353. 5057-5069 (2001)

E. Yanagida、W. M. Ni 和 P. Polacik：“影子系统中稳定解的单调性”Trans。

Y. Kubota: "Stable domains of automorphisms of Banach spaces at fixed points"Bulletin of Tokyo Gakugei University. 53. (2001)

Y. Kubota：“定点巴纳赫空间自同构的稳定域”东京学艺大学通报。