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The reaction-diffusion equation is a differential equation that mathematically models problems in various fields of the national economy such as food, chemical, biotechnology, etc., but there are still many open questions.In this book, we introduce recent developments in nonlinear reaction-diffusion equationThe book is organized in three chapters and seven sections.In Chapter 1, the mathematical theory of the initial-boundary value problem of the reaction-diffusion equation and the previous results concerning the optimal control problem of the system described by the reaction-diffusion equation are summarized, the problems to be solved in the paper are presented, and some basic results are presented.Chapter 2 demonstrates the existence and uniqueness of weak solutions to the initial-boundary value problem of a time-fractional system of reaction-diffusion equations (fractional Selkov model).In Chapter 3, we show the existence of solutions to the optimal control problem of a system described by an initial-boundary value problem of a time fractional system of reaction-diffusion equations, and derive the optimality condition of this optimal control problem.
