This paper deals with blow-up criterion for a doubly degenerate parabolic equation of the form (u^n)_t=(|u_x|^(m-1)u_x)_x+u^p in (0,1)×(0,T) subject to non- liuear boundary source (|u_x|^(m-1)u_x)(1,t)=u^q(1,t),(|u_x|^(m-1)u_x)(0,t)=0,and positive initial data u(x,0)=u_0(x),where the parameters m,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 bound-ary 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 re-gions and appropriate initial data. This extends an original work by Pablo, Quir′os and Rossi for a heat system with coupled inner absorption terms subject to homogeneous Neumann boundary conditions.
