Regional blow-up for a doubly nonlinear parabolic equation with a nonlinear boundary condition (Q2465119)

Consider the equation \((u^m)_t = (| u_x| ^{p-1}u_x)_x\), \(x\in(0,L)\), \(t>0\), complemented by an initial condition and the boundary conditions \(u_x(0,t)=0\), \(| u_x| ^{p-1}u_x(L,t)=u^\alpha(L,t)\), \(t>0\), where \(m,r,\alpha>0\) are parameters. It is known that positive solutions of this problem may blow up in finite time for some values of the parameters. The authors study the blow-up set and blow-up rate of such solutions. In particular, they find sufficient conditions on \(m,r,\alpha,L\) and the initial data guaranteeing that the blow-up is global or regional or the solution blows up in a single point.