Existence And Uniquness For Boltzmann Population Dynamics Equation In Periodic Domain

Population evolution is always change. Although there are many factors affecting it, during the change of time, the most important thing is birth, death, and transference. If we can connect them together , then we can get the mathematic equation which describes the population evolution,named population dynamics equation.The first population dynamics model was first proposed by T.R.Malthus which is a ordi-tional differential equation .However,there are some deficiencies in the model, as the research result is not consistent to actual law .Then the model has been improved by many scientists.They took into consideration of diffusion for age-structured population, and established the linear population dynamics model with diffusion.This paper deals with the generlized Boltzmann equation which was first proposed by Jager and Segel, with the aim of studying the evolution of insects. Both generlized Boltzmann equation and Boltzmann equation contain a collision operator, and both of themare the integro-differential equation of the distribution function. Bellomo and his coworkers havestudied the generlized Boltzmann equation for the spatial homogeneous and spatial inhomgeneous problems . In this article , we mainly discuss the the generlized Boltzmann equation for the spatial homogeneous problems in periodic domain .It has three parts .Firstly,we consider the linear polulation dynamics equation with diffusion .we prove the existence and uniqueness of positive weak solution with the help of trace application and fixed-point theorem.Then we prove the existence and uniqueness of positive weak solution of the the linear population dynamics equaion with integral , Further,we discuss the existence and uniqueness of positive weak solutions of regu-larization of the conditioned problem.Finally,we built the weak convergence of the loss and gain term,then dealt with the approximately equation with div-rot lemma.