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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，我们研究了一类具有热源的拟线性抛物方程：u_t-Δu^m=F in(x，t)∈Ω×(0，T)，其中m≧1，F=f(U)(通常的热源)或F=f(u(x_0(T)，t))(x_0(T))(x_0(T)∈Ω)(局部化反应)非负解的渐近行为。这个方程代表了各种现象，并给出了有趣的各种问题。(I)当m=1，Ω为有界域且F=f(u(x_0(T)，t))时，我们证明了整体解的有界性由x_0(T)的渐近行为决定，即t_→∞，这是我们关于所有解的分类结果的一部分。然而，当m&gt；1时，我们对这个问题没有很好的结果，因为我们不知道解的唯一性是否成立。(Ii)当m≧1，Ω=R^N和F=u^P时，我们研究了在空间无穷远爆破的解的精确行为。特别地，我们引入了“具有最少爆破时间的爆破解”，并证明了这样的解在空间无穷远处爆破。给出了解是具有最小爆破时间的爆破解的充要条件。我们还给出了一个爆破时间最短的爆破解在ψ方向爆破的充要条件。(3)当m&gt；1，Ω=R^N和F=u^P时，我们研究了该解在有限时间内爆破的条件，得到了新的结果。