Blow-up results for a class of first order nonlinear evolution inequalities

We investigate the non-existence of solutions to a class of evolution inequalities; in this case, as it happens in a relatively small number of blow-up studies, nonlinearities depend also on time-variable t and spatial derivatives of the unknown. The present results, which in great part do not require any assumption on the regularity of data, are completely new and shown with various applications. Some of these results referring to the problem u t = Δu + a(x) u p + λf(x) in R N , t > 0 include the non-existence results of positive global solutions obtained by Fujita and others when a ≡ 1 and f ≡ 0, Bandle-Levine and Levine-Meier when a ≡ x m and f ≡ 0, Pinsky when either f ≡ 0 or f ≠ > 0 and λ > 0, Zhang and Bandle-Levine-Zhang when a ≡ 1 and λ = 1.