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)爆破解的动力学性质一类非线性热方程的爆破解在某种弱意义下可以继续存在。又野研究动态的解决方案，从角度来看，动力系统。(2)具有随机偏差的界面运动在一类含小参数（如ε）的扩散方程中，当ε→0时，解会发展出尖锐的过渡层或界面。Funaki考虑了方程包含随机偏差项的情况。(3)Yanagida研究了一类非线性热方程的爆破现象，推广了Fujita等人的经典结果。(4)竞争系统中界面的运动Mimura研究了三种群竞争扩散系统的奇异极限中出现的界面的行为。(5)Matano推广了已有的关于对称性存在下的保序动力系统的理论。然后，他将一般理论应用于行波和其他问题的稳定性分析。

项目成果

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.

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 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.

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)

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)。