Study On Quenching Behavior Of Solutions Of Nonlinear Discrete Parabolic Equations On Graphs

With the rapid development of science and technology,nonlinear discrete parabolic partial differential equations are widely used in physics,chemistry and other fields,such as the breaking process of elastomer,the explosion caused by improper storage of lithiumion batteries,the polarization of ionic conductors and other phenomena in daily life,and the study of these phenomena finally can be transformed into the theoretical study of singular solutions of the equations.The theoretical study of singular solutions of the equations has been a hot topic for many mathematicians,and the quenching behavior of solutions of discrete parabolic equations under mixed boundary conditions is also one of the important topics.Therefore,the main research contents of this paper are as follows:In the first part,the quenching problem of the solution of discrete parabolic equations with singular absorption terms is discussed under mixed boundary conditions on graphs.The local existence of the solution is proved by Schauder’s fixed point theorem,and the discrete form of the comparison principle is given.Firstly,it is proven that the solution of the equation will undergo quenching when σ=0 and the initial values and parameters satisfy certain conditions.Secondly,when σ=0,the eigenvalue method is used to prove that the solution exists globally;when σ≠0 and the parameters meet certain conditions,it is proved that the solution of the equation will undergo quenching by constructing the upper solution of the equation.Finally,numerical simulation is used to verify the quenching behavior of the solution.In the second part,the quenching behavior of solutions of discrete parabolic equation with double singular terms on the mixed boundary condition on graphs is studied.Using the inequality expansion and contraction method,it is proved that the solution of the equation quenches in a finite time when σ=0.When σ≠0 and the parameters satisfy certain conditions,the upper solution is constructed to prove that the solution is quenched in finite time,and the upper bound of quenching time is obtained,thereby obtaining an estimate of quenching rate.The quenching phenomenon of the solution is verified by numerical simulation using the difference method.In summary,the quenching phenomena of the solutions of the above two problems are studied in this paper,and the accuracy of the theoretical results is verified by numerical simulation.The two problem models are derived from chemical catalyst dynamics or population dynamics models.Therefore,the research on the asymptotic behavior of the solution of the above problems has certain guiding significance in chemistry,biology and other aspects.