Blow-up problem for semilinear heat equation with nonlinear nonlocal   Neumann boundary condition
  In this paper, we consider a semilinear parabolic equation with nonlinear nonlocal Neumann boundary condition and nonnegative initial datum. We first prove global existence results. We then give some criteria on this problem which determine whether the solutions blow up in finite time for large or for all nontrivial initial data. Finally, we show that under certain conditions blow-up occurs only on the boundary. 
Alexander Gladkov
math.AP