Stability Of Spiky Patterns To The Singular Chemotaxis-Consumption Models

Chemotaxis is the phenomenon of directional movement of organisms caused by the stimulation of the external environment.This ability of the organism is not only of great significance to the organism itself to seek benefits and avoid harm,adapt to the environment,but also provides convenience for human daily life,such as biological decontamination,elimination of pests,treatment of tumors and so on.Therefore,chemotaxis has important application values in biology,medicine and physiology.We note that the most important feature of chemotaxis is the aggregation of organisms.From the perspective of mathematical analysis,the phenomenon of organism aggregation can be described by the finite time blow-up and spiky steady states.In this paper,we will mainly study the stability of the spiky steady states of the singular chemotaxis-consumption models in the half space.The first chapter introduces the background and research status of chemotaxis,briefly describes the characteristics of chemotaxis models and summarizes the main results of this paper.In the second chapter,we show that in the half space,when the bacterial density satisfies the zero-flux boundary condition and the nutrient concentration satisfies the non-homogeneous Dirichlet boundary condition,this model has a unique smooth spiky steady state,and it is nonlinearly stable under appropriate perturbations.The model not only has the difficulty of logarithmic singular sensitive function,but also has the difficulty of inverse square singularity of the diffusion coefficient.For the first difficulty,we use the Hopf-Cole transformation to convert the logarithmic singularity to a nonlinear nonlocality which can be solved by the method of inverse derivatives.In order to solve the second difficulty,we still maintain the structure of the original singular system and introduce a new strategy which is based on the weighted energy estimation combined with singular-weighted elliptic estimate to establish a priori estimation.In the third chapter,we consider the singular chemotaxis model with signal-dependent motility for the stability of the steady states of spikes in the half space when bacteria have zeroflux boundary conditions and oxygen has non-homogeneous Dirichlet boundary conditions.Different from the previous chapter,the model in this chapter not only has the difficulty of signal-suppressed motility,but also has the difficulty of signal-enhanced motility.We show that under the premise of small disturbance to the initial value,the spike generated by the model in the above two cases is nonlinear stable.We find that the strategies of weighted energy estimation and singular weighted elliptic estimate introduced in the previous chapter can deal with both the singularity and degeneration of chemotaxis models.The difference is,in this chapter,we need to adjust the weight several times to capture the dissipative structure of the system and the behavior of the solution near the vacuum.In the last chapter,we take the classical model as an example to study the rate at which the oxygen-consuming Keller-Segel system with a logarithmic singular sensitivity in the half space converging to a spiky steady state,where the oxygen concentration satisfies the nonhomogeneous Dirichlet boundary condition,and the bacterial density satisfies the zero-flux boundary condition.The existence and asymptotic stability of the spike have been proved previously.In this chapter,we artfully construct the weight,using the Hopf-Cole transformation combined with the weighted energy estimate to prove that the solution converges to the spike with an algebraic decay rate under appropriate initial perturbations.