Global Existence Of Weak Solutions To A Signal-Dependent Keller-Segel Model For Local Sensing Chemotaxis

Biological systems often form complex patterns or structures,such as the stripes of zebras,the stripes of leopards,and the scales of fish.The formation of the pattern of biological stripes is a question of great interest in biology.Fu and Liu proposed a kind of dynamical model based on self-trapping mechanism to study the formation of fringe pattern,which is a special form of Keller-Segel model.In this thesis,we consider global existence of weak solutions to the following chemotaxis model with the Neumann-intial boundary#12 in a smooth bounded domain with no-flux boundary conditions.The problem features a positive signal-dependent motility function ?(v)which may vanish as v becomes un-bounded which leads to the degeneracy of the first equation.In this thesis,we first modify the comparison approach to derive the upper bounds of v.Then we introduce a suitable approximation scheme which is compatible with the comparison method.By using the upper bound of v,we obtain the global existence of the classical solutions.Finally,we establish the global existence of weak solutions in any spatial dimension via compactness argument.Under the weakened assumptions on ?(v),there exists K?>0,?(v)satisfies?(v)?C3[0,?),0<?(v)?K?,(?)v?[0,?)Moreover,? satisfies 0??(K?)-1.