The extinction versus the blow-up: Global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption.
The extinction versus the blow-up: Global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption.
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DOI：
10.1016/j.jde.2017.07.029
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发表时间：
2017-11
影响因子：
2.4
通讯作者：
N. Dao;J. I. Díaz
中科院分区：
文献类型：
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作者：
N. Dao;J. I. Díaz
本文考虑具有奇异吸收项和源非线性项的退化抛物型方程∂t u−(|u x|p−2 u x)x+u−βχ{u>；0}=f(u，x，t)，在I×(0，T)上具有齐次零边界条件，I=(x1，x2)，R中的有界开区间。通过本文，我们假设p>；2和β∈(0，1).为了证明局部存在性结果，我们首先证明了|u_x|的一个精确的逐点估计。我们的主要目标之一是分析局部解可以扩展到整个时间区间t∈(0，∞)的条件，即所谓的整体解，或者相反地，出现有限时间爆破τ0>；0使得Lim t→τ0‖u(T)‖L∞(I)=+∞。此外，我们证明了，如果初始数据或源项足够小，则任何整体解在有限时间后必然相同地消失。最后，我们证明了f(0，x，t)=0，∀(x，t)∈i×(0，∞)是这类方程存在解的充要条件。
We consider nonnegative solutions of degenerate parabolic equations with a singular absorption term and a source nonlinear term:∂ t u−(| u x| p− 2 u x) x+ u− β χ {u> 0}= f (u, x, t), in I×(0, T), with the homogeneous zero boundary condition on I=(x 1, x 2), an open bounded interval in R. Through this paper, we assume that p> 2 and β∈(0, 1). To show the local existence result, we prove first a sharp pointwise estimate for| u x|. One of our main goals is to analyze conditions on which local solutions can be extended to the whole time interval t∈(0,∞), the so called global solutions, or by the contrary a finite time blow-up τ 0> 0 arises such that lim t→ τ 0‖ u (t)‖ L∞(I)=+∞. Moreover, we prove that any global solution must vanish identically after a finite time if provided that either the initial data or the source term is small enough. Finally, we show that the condition f (0, x, t)= 0,∀(x, t)∈ I×(0,∞) is a necessary and sufficient condition for the existence of solution of equations of this type.