Global Boundedness Of Solutions To A Chemotaxis-convection Model

The vascular growth of a tumour is supplied with sufficient nutrition by angiogenesis.In the process the tumour releases angiogenic factors,which stimulate endothelial cells to migrate and change into capillary sprouts.Meanwhile,the sprouts bud in a similar way.Neighbouring sprouts merge into loops at their tips.The loops may fuse with other loops,until a complex network of vessels develop.In this thesis we focus on the branching of capillary sprouts during angiogenesis and study the global bounded solution to a chemotaxis-convection model with a growth source.In Chapter 1 we briefly introduce the biological background of the tumor angiogenesis,overview the development of the related topics and give the main result.In Chapter 2 we introduce some important inequalities and related lemmas that are useful for the proof of the main result.In Chapter 3 we study a chemotaxis-convection model with a growth source(?)in a smoothly bounded domain Ω(?)R2,where d1,d2,d3,k,r,μ and δ are positive parameters,and the function f∈C1([0,∞))satisfies f(s)≤a-bsm for s>0 with a≥0 and b>0 as well as f(0)≥0.Firstly we provide the local solvability and extensibility criterion of classical solutions.Then we focus on a priori estimates to improve the regularity of classical solutions.At last,it is proved that when m>3 for all suitably regular initial data,the corresponding Neumann-type initial-boundary value problem possesses a globally defined bounded classical solution.In Chapter 4,we summarize the main contents of the thesis and look forward to some future research topics.