Study of singularities arising in nonlinear partial differential differential equations and asymptotic methods

非线性偏微分方程中奇异性的研究和渐近方法

• 批准号: 09304019
• 负责人: MATANO Hiroshi
• 金额: $ 13.76万
• 依托单位: University of Tokyo
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (A)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09304019/
• 关键词: nonlinear partial differential equations; qualitative theory; diffusion equations; infinite dimensional dynamical systems; parabolic equations; attractor; 特異性; 非線形偏微分方程式; 漸近的方法; 解の爆発; 特異極限; 界面; 拡散方程式; 無限次元力学系; 順序保存力学系; 進行波; 特異摂動問題; 自由境界 /

(1) Dynamics of blow-up solutions Some blow-up solutions of a nonlinear heat equation can be continued beyond the blow-up time in a certain weak sense. Matano studied the dynamics of such solutions from the point of view of dynamical systems.(2) Motion of interfaces with random deviation In a class of diffusion equations involving a small parameter, say ε, solutions develop sharp transition layers, or interfaces, as ε→0. Funaki considered the case where the equation involves a random deviation term.(3) Estimate of blow-up time in a nonlinear heat equation Yanagida sutdied blow-up phenomena in a nonlinear heat equation and extended the classical results of Fujita and others.(4) Motion of interface in competition systems Mimura studied the behavior of interfaces that arise in the singular limit of a three-species competition-diffusion system.(5) Order-preserving systems in the presence of symmetry Matano extended the existing theory on order-preserving dynamical systems in the presence of symmetry. He then applied the general theory to the stability analysis of traveling waves and other problems.

（1）在某种弱的意义上，可以在爆炸时间之外继续进行爆破解决方案的动力学一些非线性热方程的爆破解决方案。 Matano从动态系统的角度研究了此类溶液的动力学。（2）在涉及小参数的一类扩散方程中随机偏离的接口的运动，例如ε，解决方案，溶液会形成尖锐的过渡层或接口，为ε→0。 Funaki考虑了方程式涉及随机偏差术语的情况。（3）非线性热方程式中爆炸时间的估计Yanagida在非线性热方程中持续了爆炸现象，并扩大了藤田和其他人的经典结果。（4）竞争系统中互动系统的竞争系统的行为，竞争的行为是在竞争中竞争的5次竞争，这是一个竞争的行为。在对称Matano存在下，订单保护系统在存在对称性的情况下扩展了有关订单保护动力系统的现有理论。然后，他将一般理论应用于行进波和其他问题的稳定性分析。

项目成果

柳田 英二: "Blowup and life span of solutions for a semilinear parabolic equation"SIAM J. Math. Anal.. 29. 1434-1446 (1998)

Eiji Yanagita：“半线性抛物方程解的爆炸和寿命”SIAM J. Math.. 29. 1434-1446 (1998)

Masahiro Yamamoto: "Uniqueness and stability in multidimensional hyperbolic inverse problems"J. Math. Pures Appl.. 78. 65-98 (1999)

Masahiro Yamamoto：“多维双曲反问题的唯一性和稳定性”J.

Masahiro Yamamoto: "Uniqueness and stability in multi dimentional hyperbolic inverse problems"J. Math. Pures Appl.. 78. 65-98 (1999)

Masahiro Yamamoto：“多维双曲反问题的唯一性和稳定性”J.

三村 昌泰: "Singular perturbation problems to a combustion equation in very long cylindrical domains" Studies in Advanced Mathematics. 3. 75-84 (1997)

Masayasu Mimura：“长圆柱域中燃烧方程的奇异扰动问题”《高等数学研究》3. 75-84 (1997)。

山本 昌宏: "Uniqueness and stability in multidimensional hyperbolic inverse problems"J. Math. Pures Appl.. 78. 65-98 (1999)

Masahiro Yamamoto：“多维双曲逆问题的唯一性和稳定性”J Math. 78. 65-98 (1999)