Estimates on blow-up time of a parabolic system with nonlinear boundary flux

Abstract

A topic about a class of coupled parabolic equations under nonlinear boundary conditions is studied in this paper. Our attention is primarily on its upper and lower bounds for blow-up time. Necessary assumptions and auxiliary functions are established in the process of our proof. At the same time, we notice that the usual method to approaching the bounds is taking advantage of the different auxiliary functions. The divergence term and the cross term lead us to apply different methods to deal with the upper and lower bounds for blow-up time. Specifically, on the one hand, we seek a lower bound by taking advantage of establishing the auxiliary function and using the differential inequality technique. On the other hand, we derive an upper bound with the assistance of constructing the sub-solution and the comparison principle. In addition, two corollaries are proposed to complete our proof. Finally, we give an instance to state our results.

Keywords:

• Nonlinear parabolic system
• nonlinear boundary condition
• finite-time blow-up
• estimates of bound
• the method of sub-solution

Mathematics Subject Classifications:

• 35B44
• 35K65

Disclosure statement

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

Additional information

Funding

This paper was supported by Research Project Supported by Shanxi Scholarship Council of China (2021-030) and Key R&D Program of Shanxi Province (International Cooperation, 201903D421042).