The effect of Harnack inequality on the existence and nonexistence results for quasi-linear parabolic equations related to Caffarelli-Kohn-Nirenberg inequalities

In this work we study the problem [GRAPHICS] where Omega subset of IRN (N >= 3) is a bounded regular domain such that 0 is an element of Omega, a >= P - 1, - infinity < -gamma < N-p/p, lambda > 0, f is an element of L-1 (Omega x (0, T)) and u(0) is an element of L-1(Omega) are positive functions. The main points under analysis are some nonexistence results and complete blow-up in the case p > 2 and -gamma + 1 > 0 and some examples of existence for (gamma + 1) > 0 and 1 < p < 2. These results are interesting as they prove the role of Harnack inequality in this kind of problems and allow to understand better the blow-up behavior.