A Study Of The Transport Equation Arising In Growing Cell Populations

In this paper,using functional analysis,operator theory,semigroup theory,and other modern analytical methods,we will study a transport equation of L-R's model and Rotenberg's model arising in the growing cell populations with general boundary conditions(including integral boundary condition,local and non-local boundary conditions),and we obtain a series of new results as the constructive theory of the transport equations' solution and the spectral analysis of the corresponding transport operators and so on..The main results are following;First,for the transport equation of L-R's model arising in growing cell populations with general boundary conditions,we will study that C₀ semigroup generated by the corresponding transport operator is irreducible,moreover,we will discuss1.for the transport equation of L-R's model with integral boundary conditions in L₁ space under the conditions of the boundary operator being compact and positive, the existence of the transport operator's eigenvalue will be proved;2.for the transport equation of L-R's model with general boundaiy conditions in L_p(1≤p＜∞)space,under the conditions of the boundary operator being bounded and positive,the existence of the transport operator's eigenvalue will be proved;3.for the transport equation of L-R's model with general boundary conditions in L₂ space,under the condition of the boundary operator being bounded and positive, the existence of the transport operator's dominant eigenvalue will be proved.Second,for the transport equation of Rotenberg's model arising in growing cell populations with general boundary condition,we will study that C₀ semigroup generated by the corresponding transport operator is irreducible,moreover,we will discuss1.for the transport equation of Rotenberg's model with integral boundary conditions in L₁ space,under the conditions of the boundary operator being compact and positive,the existence of the transport operator's eigenvalue will be proved;2.for the transport equation of Rotenberg's model with general boundary conditions in L_p(1≤p＜∞)space,under the conditions of the boundary operator being bounded and positive,the existence of the transport operator's eigenvalue will be proved;3.for the transport equation of Rotenberg's model with general boundary conditions in L₂ space,under the condition of the boundary operator being bounded and positive,the existence of the transport operator's dominant eigenvalue will be proved.