| This paper mainly studies two types of models.One is the Keller-Segel model with chemotactic effect,which is mainly used to describe the collective movement of cells or microorganisms,and plays an important role in biology.The other one is the Chemotaxis-Haptotaxis model,which describes the invasion of tumor cells.It has an important guide for studying the spread of cancer cells.In this paper,we prove uniqueness of solutions to the two models by using hyper-contractivity and semigroup theory.For the keller-Segel model with chemotactic effect,we study uniqueness of the model with the Bessel potential in two-dimensional space,and existence and uniqueness of the model with Newtonian potential in higher dimensional space.The main research methods are to give the exact hyper-contractivity estimate of solutions based on the Lpestimate of solutions,and then uniqueness of solutions is proved by using the semigroup theory.There are different difficulties for different potentiasl.For the Bessel potential,the proper estimate of the Bessel kernel should be given first,and then the weak Young inequality can be used to prove uniqueness.When proving existence of solutions to the model with the Newtonian potential,it is necessary to overcome the strong singularity brought by the Newtonian potential to give a priori estimate,and then obtain existence of solutions by regularizing model and making compactness discussion.For the Chemotaxis-Haptotaxis model in whole space,the uniqueness of solutions is studied for the models without and with logistic sources,respectively.The main difficulty is to deal with the influence of the Haptotaxis term in the model.So we need to estimate the Haptotaxis term in the model first,then prove uniqueness of solutions.Through the study of the above two kinds of models,it can be found that the study of uniqueness of solutions to the model with semigroup structure can be realized by using hyper-contractivity combined with semigroup theory. |
PDF Full Text Request