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.

在这项研究中，我们调查了具有功率非限度的半线性扩散方程的爆炸问题。考虑了在Neumann边界条件下方程的爆炸速率或非凸线域中的Dirichlet边界条件。由于Giga Kohn导致的良好结果不能应用于这些情况下，因此我们表明爆炸是使用Liouville型定理的I型。在这里，如果爆炸速率与相应普通方程的解决方案相同，则表示该方程的解决方案表现出I型爆炸。接下来，在大扩散的情况下，该域的特性描述了库奇 - 尼曼问题的爆炸解决方案的1个位置。我们还研究了Sobolev意义上超临界指数的爆炸后的延续。解决方案在有限的时间内爆炸，并在爆炸后以各种行为倾向于无穷大。此外，我们得到了一个解决方案，该解决方案在有限的时间内炸毁，并在爆炸时间后立即变为常规，然后再次炸毁。

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