Study on the asymptotic behavior of solutions of quasilinear parabolic equations with a blow-up term

带爆炸项的拟线性抛物型方程解的渐近行为研究

基本信息

批准号：
17540171
负责人：
SUZUKI Ryuichi
金额：
$ 2.12万
依托单位：
Kokushikan University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2007
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-17540171/
关键词：
parabolic quasilinear semilinear blow-up the Cauchy problem the Dirichlet problem localized reaction space infinity 大域解非存在解の爆発空間無限遠有界性

项目摘要

In our project, we study the asymptotic behavior of nonnegative solutions of the Dirichlet problem(Ω is bounded) or the Cauchy problem(Ω = R^N) for a quasilinear parabolic equation with a heat source : u_t-Δu^m= F in(x, t)∈ Ω ×(0, T), where m ≧1, and F =f(u)(a usual heat source) or F = f(u(x_0(t), t))(x_0(t)∈Ω)(localized reaction).Furthermore, we assume that f satisfies some blow-up condition. This equation represents various phenomena and gives interesting various problems. We have obtained the next three results for these problems.(i) When m=1, Ω is a bounded domain and F=f(u(x_0(t), t)), we showed that the boundedness of global solutions is determined by the asymptotic behavior of x_0(t)as t→∞.This result is a part of our result on the classification of all solutions. However, when m> 1, we do not have good results for this problem, since we do not know whether or not the uniqueness of solutions holds.(ii)When m ≧1, Ω= R^N and F=u^P , we studied the precise behavior of solutions which blow up at space infinity. In particular, we introduced "blow-up solution with the least blow-up time" and showed that such a solution blows up at space infinity. We give a necessary and sufficient condition for a solution to be a blow-up solution with the least blow-up time. We also give a necessary and sufficient condition for a blow-up solution with the least blow-up time to blow up in a direction ψ.(iii)When m>1, Ω= R^N and F = u^P , we studied under what condition the solution blows up in finite time, and got new results.

在我们的项目中，我们研究具有热源的拟线性抛物型方程的狄利克雷问题（Ω 有界）或柯西问题（Ω = R^N）非负解的渐近行为： u_t-Δu^m= F in (x, t)ε Ω ×(0, T)，其中 m ≥1，并且 F =f(u)（通常的热源）或 F = f(u(x_0(t))， t))(x_0(t)εΩ)(局部反应)。此外，我们假设 f 满足一些爆炸条件。这个方程代表了各种现象，并给出了这些问题的接下来的三个结果。 .(i) 当 m=1，Ω 是有界域且 F=f(u(x_0(t), t)) 时，我们证明全局解的有界性由x_0(t)as t→∞。这个结果是我们对所有解的分类结果的一部分。但是，当 m> 1 时，我们对此问题没有好的结果，因为我们不知道是否解的唯一性成立。(ii)当 m ≥ 1、Ω= R^N 且 F=u^P 时，我们研究了在无限远空间爆炸的解的精确行为，特别是，我们引入了“爆炸解”。最少的爆炸时间”，并表明这样的解在无限远空间爆炸。我们给出了一个解成为具有最短爆炸时间的爆炸解的充要条件。我们还给出了一个充要条件对于在 ψ 方向上爆炸所需的最短爆炸时间的爆炸解。(iii)当 m>1、Ω= R^N 且 F = u^P 时，我们研究了溶液在什么条件下爆炸在有限的时间内，得到了新的成果。