In the course of an investigation of the nonlinear Fourier equation of heat conduction by a method of analytic continuation with respect to the temperature maximum y m as the continuation parameter, it is shown how positive majorants and positive minorants (which are defined to vary with the parameter y m ) can be used to furnish continuous upper and lower bounds of a function λ( y m ) which governs the intensity of the generation of heat in the course of a certain exothermic process. The function λ( y m ) constitutes a characteristic property of the problem because it contains information as to the existence of stable, unstable and critical temperature distributions which result from the boundary-value problem of Fourier's equation. In order to provide a test for the proposed method, continuous upper and lower bounds of λ( y m ) are derived for a Fourier problem of which the exact numerical solution is known.