About the absence of global solutions to nonlinear evolutionary differential equations and inequalities

There are considered generalized solutions of evolutionary inequalities for an arbitrary order both in time and in spatial variables. Exact conditions have been established for structure of inequalities that guarantee the destruction of an arbitrary local solution of such inequalities. In the case of inequalities of the first order in time and second in space variable, these conditions are a new exact form of the Fujita-Hayakawa conditions, in the case second-order inequalities in time and in spatial variables — an exact analogue of Kato conditions.