Global existence and blow-up for a fourth order parabolic equation involving the Hessian
Global existence and blow-up for a fourth order parabolic equation involving the Hessian
复制标题
涉及 Hessian 矩阵的四阶抛物型方程的全局存在性和爆炸
DOI：
10.1007/s00030-017-0465-7
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Jun Zhou
中科院分区：
文献类型：
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作者：
Guangyu Xu;Jun Zhou
研究了Escudero等人（J Math Pures Appl 103（4）：924-957，2015）最近研究的一类含Hessian的四阶抛物型方程，得到了当初始能量为山口水平时，方程-范数和-范数爆破的初始条件.本文的目的是研究本文中提出的两个公开问题，即-范数爆破和解的行为。对于的情形，我们证明了解在有限时间内以-范数爆破。此外，我们估计了爆破时间和爆破率。对于的情形，我们找到了两个集合，并证明了当初值属于时，解在有限时间内爆破，而当初值属于时，解整体存在并随时间趋于零。
This paper deals with a fourth order parabolic equation involving the Hessian, which was studied in Escudero et al. (J Math Pures Appl 103(4):924–957, 2015) recently, where the initial conditions for-norm and-norm blow-up were got when the initial energy, whereis the mountain-pass level. The purpose of this paper is to study two of the open questions proposed in the paper, that is,-norm blow-up and the behavior of the solutions when. For the case of, we prove the solution blows up in finite time with-norm. Moreover, we estimate the blow-up time and the blow-up rate. For the case of, we find two setsand, and prove that the solution blows up in finite time if the initial value belongs to, while the solution exists globally and tends to zero as timewhen the initial value belongs to.