The paper addresses the very popular issue of diffusion of innovations. Diffusion of innovations is the last stage of the innovation process and an indicator of its success. The study of the diffusion of innovations and the application of heat conduction equations in this theory is related to the study of innovative characteristics and economic indicators that could improve the performance of enterprises, lead to profit growth and expand market coverage. The paper systematized the factors that influence the diffusion of innovations. With their help, it is possible to manage the process of diffusion of innovations in a specific market and in a certain period of time. The use of the theory and methods for solving the thermal conductivity equation allows obtaining a solution to the problem of the diffusion equation, finding the optimal parameters for creating an innovative product susceptibility on the market.

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In this paper, we propose a Supplement to the theory of the emergence of innovations in long Kondratiev cycles. The regularities of the emergence of crisis phenomena and the contribution of J. Schumpeter to the development of the innovative economy are considered. A comparative analysis of Marketing strategies is carried out. Schumpeter and N. D. Kondratiev, identified distinctive factors and common combinations. The stages of long wave cycles over several decades are analyzed. The analysis revealed innovative peaks that occur in the middle of the depression phase, followed by stages of economic activity growth after a certain period of time. The authors studied the theory of the trigger effect of depression, similar to the trigger action, which results in an avalanche of innovation growth. To confirm the theory of the existence of the “trigger effect of depression”, statistical data on the number of patents and applications filed in Russia and the United States for several decades, which are indicators of innovation activity, are presented. The authors put forward a hypothesis about the possibility of describing the periodic change in the number of innovations over time using “parametric resonance” models and the Mathieu equation.