A duality principle corresponding to the parabolic equations

Our principal problem consists in finding threshold values for a family of blow-up solutions of nonlinear evolution problems depending on a real parameter λλ. We seek to obtain the solution of the problem in a constructive and unified approach which would provide the answer for the following general question: Is there is a constructive method for finding the threshold values? We illustrate our approach by means of an example of finding a threshold value for a family of parabolic equations with nonlinearity indefinite in sign. In the main result, the threshold value is expressed via explicitly specified dual variational principles of a new type.