This paper deals with blow-up criterion for a doubly degenerate parabolic equation of the form （u^n）t = （｜ux｜^m-1ux）x ＋ u^p in （0, 1） × （0,T） subject to nonlinear boundary source （｜ux｜^m-1ux）（1,t） = u^q（1,t）, （｜ux｜^m-1ux）（0,t） = O, and positive initial data u（x,0） = uo（x）, where the parameters va, n, p, q 〉0. It is proved that the problem possesses global solutions if and only if p ≤ n and q〈min {n,m（n＋1）/m＋1}.
We study finite time quenching for heat equations coupled via singular nonlinear boundary flux. A criterion is proposed to identify the simultaneous and non-simultaneous quenchings. In particular, three kinds of simultaneous quenching rates are obtained for different nonlinear exponent regions and appropriate initial data. This extends an original work by Pablo, Quirós and Rossi for a heat system with coupled inner absorption terms subject to homogeneous Neumann boundary conditions.
