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

In this work, positive solutions to a doubly nonlinear parabolic equation with a nonlinear boundary condition are considered. We study the problem {(u(m))(t) = (vertical bar u(x)vertical bar r(-1) u(x))(x), (x, t) epsilon (0, L) x (0, T-max), {u(x) (0, t) = 0, vertical bar ux vertical bar(r-1) u(x)(L, t) = u(alpha)(L, t), t epsilon (T-max), {u (x, 0) = phi(x), x epsilon [0, L], where 0 < m, r, alpha < infinity are parameters. It is known that for some values of the parameters there are solutions that blow up in finite time. We determine in terms of alpha, m, r the blow-up sets for these solutions. We prove that single point blow-up occurs if max{m, r} < alpha, global blow-up appears for the range of parameters 0 < m < alpha < r and regional blow-Lip takes place if 0 < m < alpha < r and L>L* equivalent to r + 1/ r - m. In this case the blow-up set consists of the interval [L - L*, L] subset of (0, L].