This paper deals with a pseudo-parabolic equation involving $p\left(x\right)$-Laplacian, which has been studied in Di et al. (2017), Zhu et al. (2020) and Liao (2019). By the variational methods and embedding inequalities, we determine the complete classification of blow-up and global existence of weak solutions in some Sobolev space with variable exponent by covering all kinds of initial energy characterized with the mountain pass level.

中文翻译：

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初始能量的完整分类 $p（X）$-Laplace伪抛物方程

本文涉及一个伪抛物线方程 $p（X）$-Laplacian，已在Di等人中进行了研究。（2017），Zhu等。（2020）和廖（2019）。通过变分方法和嵌入不等式，我们涵盖了具有山口水准的各种初始能量，从而确定了具有可变指数的Sobolev空间中爆破和弱解的整体存在的完整分类。