Asymptotic analysis for a pseudo-parabolic equation with nonstandard growth conditions

Abstract

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.

COMMUNICATED BY:

• X. Zou

Keywords:

• Pseudo-parabolic equation
• nonstandard growth conditions
• potential well method
• asymptotic behavior

2010 Mathematics subject classifications:

• Primary 35K58
• 35K35
• 35B40

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (11401078, 11571062, 11701067, 11871134).