Study On Chemotaxis Models With Nonlinear Signal Secretion Mechanism

This paper mainly focuses on two kinds of chemotaxis models with nonlinear signal secretion mechanism.Firstly,we study the homogeneous Neumann initial boundary value problems of the following chemotaxis model with indirect nonlinear chemical signal secretion:where ?(?)Rn(n>2)is a bounded domain with smooth boundary,?? 0 f(s)?C1([0,+?))satisfying 0?f s)?ks? for all s>0 and some positive constants k and ?.When ?=0 and 0<?<2/n,the global existence and boundedness of this initial boundary value problem is established by using the constant variation method,semigroup theory,and the regularization properties of inhomogeneous linear heat equation.Secondly,we study the homogeneous Neumann initial boundary value problem of the following chemotaxis model with two nonlinear chemical signal secretion:where ?(?)R2 is a bounded domain with smooth boundary,function f(s)?C1([0,+?))satisfies 0?f(s)?Kts?((?)s?0),and function g(s)? C([0,+?))satisfies 0?g(s)?Kgs?((?)s?0),where Kd,Kg,?,? are positive constants.By using the constant variation method,semigroup theory,regularization property of inhomogeneous linear heat equation,Moser iteration and other methods,it is shown that when ?>0,if min{?,?}<1,for arbitrarily large initial data,the model has a unique global classical solution which is uniformly bounded on ?×(0,?).When ?=0,if??<1,similar result is also validFinally,we give a brief summary on the main contexts of this thesis,and provides several problems which will be studied in the future.