The Asymptotic Behavior Of The Solution To Two Types Of The Discrete Equation With Nonlinear Term

With the development of science and technology,the discrete equations are widely applied in a variety of fields in our society,such as energy transformation,electrical network and image processing.Therefore,based on the actual demands of production and life,the discrete heat equation has also been concerned and studied by many scholars at domestic and abroad,However,it is not perfect for the study of the properties of the solution to the discrete heat equation with nonlinear terms.Thus,this paper will discuss the asymptotic behavior solution to heat equations with other types of nonlinear terms.This paper is divided into the following four chapters:In the first chapter:We introduce the basic concepts which were used in this article.In the second chapter:We mainly study the relations between the solution of the discrete Poisson equation and the solution of the discrete heat equation with exponential nonlinear term by monotone iterative method and comparison principle.When the solutions of the discrete equation exist,we discuss the asymptotic stability of the solutions to the discrete heat equations with exponential nonlinear term.In the third chapter:We mainly consider blow-up for the discrete heat equation with exponential reaction term on finite graph.First,we study its blow-up time and blow-up rate by comparison principal.And then,we prove that there exists a critical exponent~*?such that the solution blows up in finite time,when??~*?,and,the problem admits a global solution all??~*?by the Implicit Function Theorem.Finally,some numerical experiments illustrate the theoretical results.In the last chapter:we mainly consider quenching of the solution to the discrete heat equation with logarithmic type sources on graphs,first,The local existence and uniqueness of solutions are obtained by Banach fixed point theorem.And then,the solution quenches in finite time and the blow-up of its time-derivatives at the quenching time is verified by the comparison principle under some suitable conditions.On the other hand,we also prove that there exists a critical exponent~*?,when???~*,the above problem admits a global solution by the Implicit Function Theorem.Moreover,when??~*?,its solution will quench at finite time.Finally,some numerical experiments are used to illustrate the theoretical results.