Research On The Properties Of Solutions For Cauchy Problem In Two-dimensional Chemotaxis-haptotaxis Model

Chemotaxis-haptotaxis model,as a special chemotaxis model,plays an important role in biomathematics.The model describes the process of cancer invasion by studying the interaction among cancer cells,matrix degrading enzymes and extracellular matrix.Due to its practical application background and rich mathematical logic,this model has attracted the attention of many researchers in recent years.This paper mainly studies the properties the solutions of the Cauchy problem to chemotaxis-haptotaxis models in R~2,including the optimal initial criterion of the solutions to the chemotaxis-haptotaxis term model,the existence and L^?boundedness of solutions to a chemotaxis-haptotaxis model with logistic source.Specifically,for the chemotactic-haptotaxis model without logistic source,it is proved that the critical mass of solutions of the model is 8?.That is,when the initial mass is less than 8?,solutions of the model globally exist?when the initial mass is greater than8?and the initial second moment is small enough,the solutions blow up in a finite time.The best initial mass mainly comes from the construction of a proper free energy and the applica-tion of the Brezis-Merle inequality with the best constant.For the chemotaxis-haptotaxis model with logistic source,we give the global existence of solutions by constructing the approxima-tion problem,making uniform estimation and compactness discussion,and further using Moser iterative technique proves L^?boundedness of solutions.Compared with the existing literature,this paper obtain the optimal criterion of solutions of the Cauchy problem to chemotaxis-haptotaxis models first,and it gives that the critical mass of solutions of the model is 8?.It is the same as the critical mass of the classical Keller-Segel model,which shows that the weak concentration effect of the haptotaxis term has no essential effect on the critical mass of solutions.At the same time,the global existence and L^?boundedness of solutions under the condition of large initial datas are obtained for the chemotaxis-haptotaxis model with logistic.By comparing the results of two kinds of chemotaxis-haptotaxis models with or without logistic source,it is found that the existence of logistic source will prevent the blow-up of solutions.