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

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.