The Blow-up Phenomenon Of Reaction-diffusion Equation And Its Application In Epidemic Model

The reaction-diffusion equations are mainly used to study the spatial distribution and diffusion law of natural systems,analyze the impact of time and space on the systems’diffusion,so as to more accurately grasp the impact of the diffusion rate on the surrounding environment.Many nonlinear phenomena in the field of biomedical engineering can be described by reaction diffusion equations,such as:the interaction of biomolecules and cells and the law of cell growth,the occurrence,spread and devel-opment trend of infectious diseases.This paper focuses on the blow-up phenomenon for reaction-diffusion equations,and their some applications in epidemic model.In theory,by means of auxiliary functions methods,Sobolev space theory,maximum prin-ciple,first-order differential inequality techniques,etc.,we study the blow-up problem for several types of reaction-diffusion equations,and obtain the results about the exis-tence of the global solution and the blow-up solution and the estimation of the upper and lower bounds of the blow-up time.The related conclusions in the literature are generalized and improved;in application,based on the SIS epidemic model with satu-rated incidence,two types of reaction-diffusion models are established,and the dynamic behaviors of the models are studied.Moreover,the main conclusions are verified by numerical simulation.The main contents of the paper are described as follows:First,blow-up problems for several types of reaction-diffusion equations under different boundary conditions are discussed.By constructing appropriate auxiliary functions and applying the technique of first-order differential inequality,the existence of the blow-up solution for reaction-diffusion equation with the nonlinear boundary conditions is investigated,and the upper and lower bound estimates of the blow-up time are given;With the help of constructing appropriate auxiliary function and H（?）lder inequality,Young inequality and Poincar（?） inequality,we investigate sufficient conditions not to blow up for the solution of the reaction-diffusion equation with Robin boundary conditions,that is,the conditions for the existence of the global solution.Moreover,combining the first-order differential inequality with the maximum principle,we prove that the conditions to guarantee the solution blows up.When the blow up occurs,the upper and lower bound estimates of the blow-up solution are given;The reaction-diffusion equation with non-local boundary conditions is studied.Through appropriate auxiliary functions and differential inequality techniques,we prove the existence for blow-up solution under different auxiliary functions,and the upper bound estimate of the blow-up time is obtained.In addition,we respectively give the lower bound estimate of blow-up time in R³or higher dimensional space R⁹)（9）≥3).Second,the blow-up problem of coupled reaction-diffusion equations with non-local boundary conditions is investigated.And we obtain sufficient conditions for existence of blow-up solution and the upper bound estimate of the blow-up time.In addition,applying some inequalities,such as Sobolev inequality,we give the lower bound estimate of the blow-up time.We enrich relevant results.Finally,the applications of reaction-diffusion equation in the epidemic model are discussed,and the dynamic behaviors of two types of SIS reaction-diffusion epidemic model with saturation incidence are studied.We prove the boundedness of the solution and the uniqueness of the disease-free equilibrium point for the SIS reaction-diffusion epidemic model with saturation incidence and Logistic source.The stability of the disease-free equilibrium point and the global attractiveness of the endemic disease bal-ance point are studied,and the main conclusions and the impact of saturation on the disease are verified through numerical simulations;At the same time,the dynamic behavior of the SIS reaction-diffusion epidemic model with saturation incidence and saturation treatment is studied.We not only prove the existence and uniqueness of the disease-free equilibrium point of the model,but also proves the stability of the disease-free equilibrium point based on the basic reproduction number.By utilizing the method of the upper and lower solution,we study the existence of the endemic dis-ease equilibrium point.Moreover,the main conclusions and the impact of saturation treatment on the disease are verified by numerical simulation.