Large-time Behaviors Of Solutions Of Transportation Network Models

Biological transportation networks,such as blood vessels,leaf venation in plants and neural networks,are fundamental components in living systems.To better understand the formation and evolution of the transportation networks,the scientists have established many mathematical models by using partial differential equations.The study of the math-ematical properties of these models has become a hot topic in the current study of partial differential equations.This dissertation mainly studies the properties of solutions of two kinds of partial differential equations for biological transport network models,such as existence,blow-up criteria,uniform boundedness and large-time behavior,etc.The main contents and results are as follows:1.We study the Cauchy problem of a three-dimensional fluid transportation network model.Firstly,we establish the local existence of strong solutions and present a blow-up criterion;secondly,we show that the solutions exist globally and are uniformly bounded under the some smallness conditions of initial data.2.We study the initial-boundary value problem of the fluid transportation network model.Firstly,in the two and three dimensional settings,we show that the classical small-data solution is uniformly bounded by employing the elliptic estimates as well as trading time derivative and spacial derivative.Secondly,in the one dimensional setting,by means of the uniform boundedness of weak large-data solution we prove that the solution van-ishes in finite or infinite time.Thirdly,invoking the uniform boundedness of classical small-data solution,the same phenomena are presented in the two and three dimensional settings.3.We explore the effects of diffusion coefficient and activation parameter on the properties of solutions of fluid transport network model.Firstly,we obtain the global existence and time-decay estimates of classical-small solutions for the initial-boundary value problem of the reduced system.Secondly,when the diffusivity D~2tends to infinite,the convergent rate of the solution of the original initial-boundary value problem toward the solution of the reduced initial-boundary value problem is obtained.4.We study the initial-boundary value problems for a class of ion transport network models.Specifically,under the some smallness conditions of initial data,by employing a-priori decay assumption and the continuation argument we obtain the global existence,uniform boundedness as well as the time-decay rate of the solution.