Free Boundary Problems Of Some Reaction-diffusion Equations

In this doctoral thesis, we focus on some free boundary problems of ecological mod-els to describe the spreading of invasive species in their new habitats, which include a Lotka-Volterra prey-predator model with a invasive predator, a Lotka-Volterra competi-tive model with two invasive species, two ecological models with stage structure and a invasive species.At first, we study two free boundary problems of a Lotka-Volterra prey-predator model in one dimension space and in radially symmetric environment respectively. The existence and uniqueness of local solution are proved by virtue of LP theory for parabolic equation and contraction mapping theorem, and the local solution can be extended to all t>0. We construct a exquisite functional sequence and a specially iterative format to get a spreading-vanishing dichotomy for the invasive predator. Moreover, establishing comparison principle and constructing a suitable supper solution, we give some sufficient conditions for spreading or vanishing of the invasive predator.Then, we consider a free boundary problem of Lotka-Volterra competitive model with two invasive species. The main objective is to understand the dynamics of the two competitive species spreading via a free boundary. We construct a suitable functional sequence and a specially iterative format to obtain a spreading-vanishing dichotomy for the two species. That is to say, either at least one species successfully spreads to the right-half-space as time t goes to infinity, or the two species fail to establish and die out in the long run. And the dynamics of such a problem are same as those of the correspond-ing ODE system. Moreover, some sufficient conditions are also given for spreading or vanishing of the two species.Finally, we investigate two free boundary problems of ecological models with stage structure, which are one species model and two species competitive model. For the one species model, suitable supper-lower solutions are constructed to gain the asymptotic behavior of the solution for the corresponding initial problem. Then a spreading-vanishing dichotomy is obtained for the free boundary problem. Moreover, choosing the expanding coefficient of the free boundary as a parameter, we establish comparison principle and construct a suitable supper solution to show the existence of the sharp criteria governing spreading and vanishing. For the competitive model, on the basis of the results obtained in the one species model, we sandwich the free boundary problem in between two auxiliary initial-boundary problems with fixed boundary, obtain the vanishing result or a spreading-vanishing dichotomy of the invasive species under different conditions, and give the sharp criteria governing spreading and vanishing.