Complete and energy blow-up in parabolic problems with nonlinear boundary conditions

We study the possible continuation of solutions of a nonlinear parabolic problem after the blow-up time. The nonlinearity in the equation is dissipative and blow-up is caused by the nonlinear boundary condition of the form partial derivative u/partial derivative v = vertical bar u vertical bar(q-1) u, where q > 1 is subcritical in H-1 (Omega). If the dissipative term in the equation is linear then we show that blow-up of positive solutions is complete. If the dissipative term is superlinear then the solution can be continued inside the spatial domain. On the other hand, we find sufficient conditions on the nonlinearifies guaranteeing that no reasonable continuation can be expected on the boundary. (c) 2005 Elsevier Ltd. All rights reserved.