Some remarks on quasilinear parabolic problems with singular potential and a reaction term

In this paper we deal with the following quasilinear parabolic problem {(u(theta))t - Delta pu = lambda u(p-1)/vertical bar x vertical bar(p) + u(q) + f, u >= 0 in Omega x (0, T), u(x, t) = 0 on partial derivative Omega x (0, T), u(x, 0) = u(0)(x), on x is an element of Omega, where theta is either 1 or (p - 1), is either a bounded regular domain containing the origin or , 1 < p < N, q > 0 and u 0 a parts per thousand yen 0, f a parts per thousand yen 0 with suitable hypotheses. The aim of this work is to get natural conditions to show the existence or the nonexistence of nonnegative solutions. In the case of nonexistence result, we analyze blow-up phenomena for approximated problems in connection with the classical Harnack inequality, in the Moser sense, more precisely in connection with a strong maximum principle. We also study when finite time extinction (1 < p < 2) and finite speed propagation (p > 2) occur related to the reaction power.