On the analysis of blowup phenomena for a nonlinear parabolic equation

非线性抛物方程的爆炸现象分析

• 批准号: 15540199
• 负责人: MIZOGUCHI Noriko
• 金额: $ 2.3万
• 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: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2004
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540199/
• 关键词: semilinear heat equation; blowup; incomplete blowup; continuation of solution; backward self-similar solution; 半線形放物型方程式; 爆発集合; Neumann境界条件; supercritical; 逆向き自己相似

In this research, we investigated a blowup problem for a semilinear diffusion equation with power nonlinerity. The blowup rate for the equation under the Neumann boundary condition or the Dirichlet boundary condition in a non-convex domain is considered. A well-lnown result due to Giga Kohn cannot be applied to these cases, so we showed the blowup is of type I making use of Liouville type theorem. Here a solution of the equation is said to exhibit the type I blowup if the blowup rate is as same as that of solutions to the corresponding ordinary equation. Next, the 1 location of the blowup set of solutions to the Cauchy-Neumann problem is described by the property of the domain in the case of large diffusion. We also studied the continuation after blowup for supercritical exponent in the Sobolev sense. Solutions blowing up in finite time and being a classical solution for all time after blowup with various behaviors as time tends to infinity. Moreover, we got a solution which blows up in finite time and becomes regular immediately after the blowup time and blows up again.

本文研究了一类具有幂非线性项的半线性扩散方程的爆破问题。研究了非凸区域上方程在Neumann边界条件和Dirichlet边界条件下的爆破速率。由于Giga Kohn的一个著名结果不能应用于这些情形，所以我们利用Liouville型定理证明了爆破是I型的.这里，如果爆破速率与相应常方程的解的爆破速率相同，则称方程的解表现出I型爆破。其次，在大扩散的情况下，Cauchy-Neumann问题解的爆破集的1位置由区域的性质来描述。我们还研究了Sobolev意义下超临界指数爆破后的延拓问题。解在有限时间内爆破，爆破后的所有时间都是经典解，并且随着时间趋于无穷大，解的行为会发生变化。此外，我们还得到了一个解，它在有限时间内爆破，在爆破后立即正则化，然后再次爆破。

项目成果

Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition

• DOI: 10.1016/s0022-0396(03)00128-1
• 发表时间: 2003-09
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: N. Mizoguchi

N.Mizoguchi: "Blowup rate of solutions for a semilinear heat equation with the Neumann boundary condition"J.Differential Equations. 193. 212-238 (2003)

N.Mizoguchi：“具有诺依曼边界条件的半线性热方程解的爆炸率”J.微分方程。

A Liouville property and quasiconvergence for a semilinear heat equation

• DOI: 10.1016/j.jde.2003.10.019
• 发表时间: 2005
• 期刊: Journal of Differential Equations
• 影响因子: 2.4
• 作者: P. Polácik;E. Yanagida

N.Mizoguchi: "Blowup rate of solutions for a semilinear heat equation with the Dirichlet boundary condition"Asymptotic Analysis. 35. 91-112 (2003)

N.Mizoguchi：“具有狄利克雷边界条件的半线性热方程解的爆炸率”渐近分析。

Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation

• DOI: 10.57262/die/1356060346
• 发表时间: 2004-01
• 期刊: Differential and Integral Equations
• 影响因子: 1.4
• 作者: P. Polácik;E. Yanagida