Blow‐up for a class of nonlinear parabolic problems

For nonlinear parabolic equations of the form u_t=Δu^m−u^μ‖∇u^m‖^q+u^p, we prove nonexistence of global admissible solutions for large initial data for some range of the parameters m, μ, q and p. To do so we use comparison with suitable blowing up self‐similar subsolutions. We also prove that for the complementary range of the parameters for which we obtain blow‐up, there exists globally bounded admissible solutions.