Existence and nonexistence results for a class of parabolic equations with mixed boundary conditions

We study the following parabolic problem [GRAPHICS] where Omega subset of R-N is a smooth bounded domain with 0 is an element of Omega, B(u) = u chi(Sigma 1 x (0,T)) + vertical bar x vertical bar(-p gamma)vertical bar del u vertical bar(p-2)partial derivative u/partial derivative v chi(Sigma 2 x (0,T)) and -infinity < gamma < N-p/p. The boundary conditions over partial derivative Omega x (0, T) verify hypotheses that will be precised in each case. Mainly, we will consider the second member f(x,u) = u(alpha)/vertical bar x vertical bar(p(gamma+1)) with alpha >= p-1, as a model case. The main points under analysis are some existence, nonexistence and complete blow-up results related to some Hardy-Sobolev inequalities and a weak version of Harnack inequality, that holds for p >= 2 and gamma + 1 > 0.