论文标题
将爆炸和全球存在的解决方案分类为初始$ \ textrm {neumann} $问题
Classification of blow-up and global existence of solutions to an initial $\textrm{Neumann}$ problem
论文作者
论文摘要
本文的目的是应用修改后的潜在井方法和一些新的差异不平等，以研究解决方案的渐近行为对最初均匀的$ \ hbox {neumann} $问题的问题是由$ p（x）$ - \ hbox {laplace} operator驱动的非线性扩散方程的问题。当初始数据满足不同的条件时，在有限的解决方案的有限时间内对全球存在和爆炸的完整分类。粗略地说，我们获得了一个阈值结果，使解决方案在全球存在或在初始能量分别是亚临界和关键的有限时间内爆炸。此外，还为全球解决方案获得了$ l^2 $规范的衰减率。还为超临界初始能量提供了足够的全球和爆破解决方案的条件。最后，当扩散项主导源时，我们给出了渐近行为的双向估计。这是我们以前的工作\ cite {gg}的延续。
The aim of this paper is to apply the modified potential well method and some new differential inequalities to study the asymptotic behavior of solutions to the initial homogeneous $\hbox{Neumann}$ problem of a nonlinear diffusion equation driven by the $p(x)$-\hbox{Laplace} operator. Complete classification of global existence and blow-up in finite time of solutions is given when the initial data satisfies different conditions. Roughly speaking, we obtain a threshold result for the solution to exist globally or to blow up in finite time when the initial energy is subcritical and critical, respectively. Further, the decay rate of the $L^2$ norm is also obtained for global solutions. Sufficient conditions for the existence of global and blow-up solutions are also provided for supercritical initial energy. At last, we give two-sided estimates of asymptotic behavior when the diffusion term dominates the source. This is a continuation of our previous work \cite{GG}.
