The main goal of this work is to study a semilinear pseudo-parabolic equation with nonstandard growth conditions. For initial energy $J\left(u_0\right)\le d$, by using the generalized potential well method, we establish a sharp threshold result on global existence or blow-up weak solution. On the other hand, we also show the asymptotic behavior of solutions and the blow-up time estimates. The optimal estimations: the asymptotic behavior are obtained when the blow-up occurs. Moreover, for high initial energy $J\left(u_0\right)>d$, we illustrate the conditions on global existence or blow-up.

中文翻译：

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非标准生长条件下拟抛物线方程的渐近分析

这项工作的主要目标是研究具有非标准生长条件的半线性伪抛物线方程。对于初始能量$Ĵ\left({你}_0\right)\le d$，通过使用广义势阱方法，我们建立了全局存在或爆破弱解的尖锐阈值结果。另一方面，我们还展示了解的渐近行为和爆炸时间估计。最优估计：在发生爆破时获得渐近行为。此外，对于高初始能量$Ĵ\left({你}_0\right)>d$，我们说明了全球存在或爆炸的条件。