Blowup property of solutions in the parabolic equation with p-Laplacian operator and multi-nonlinearities

ABSTRACT

In this paper, we study blowup properties of weak solutions in their $W^{1,\mathrm{\infty }}$ norm to the degenerate parabolic equation with multi-nonlinearities and gradient terms. First, we show the existence and uniqueness of weak solutions by using the priori estimate methods. Second, we obtain the global existence criteria and blowup criteria after proving some gradient estimates for different coefficients. Third, we use some Sobolev's inequalities and deal with some differential inequalities of new barrier functions to determine some upper and lower bounds of blowup time of solutions. It could be found out that the blowup or global existence phenomena depend sensitively on the relationship among the different exponents of nonlinearities, which discover a key clue to the different effect of diffusion, gradient term, source term and absorption term on the singular properties of weak solutions.

KEYWORDS:

• $L^{\mathrm{\infty }}$ blowup
• gradient blowup
• global existence
• blowup time

MSC (2010):

• 35K55
• 35B40
• 35K15
• 35B33

Acknowledgments

The authors would like to express their sincerely thanks to the Editors and the Reviewers for the constructive comments to improve this paper.

Data availability statement

All data generated or analysed during this study are included in this article.

Disclosure statement

The authors declared that they have no conflicts of interest to this work.

Additional information

Funding

This paper is supported by Shandong Provincial Natural Science Foundation of China (ZR2021MA003,ZR2020MA020) and the Fundamental Research Funds for the Central Universities (202111015).