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

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。

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.