On the critical Fujita exponents for solutions of quasilinear parabolic inequalities

This work is devoted to the study of critical blow-up phenomena for wide classes of quasilinear parabolic equations and inequalities. The model example for this treatment is well known and comes from the theory of turbulent diffusion: ut divx |u|m−1u x α−2 |u|m−1u x + |u|q−1u, m 1, α>1. (∗) We obtain the existence of critical blow-up exponents for nonnegative weak solutions of inequalities of the type (∗) that belong only locally to the corresponding Sobolev spaces in the half-space R1 + × Rn, n 1. We impose no conditions on the behavior of solutions on the hyperplane t = 0 so that the results obtained in the paper are of Liouville type. The approach developed herein is directly applicable to the study of analogous problems for systems of quasilinear elliptic and parabolic equations and inequalities, and its elliptic analogue has recently been developed by the authors.  2002 Elsevier Science (USA). All rights reserved.