Some Kinds Of Free Boundary Problems With Local Or Nonlocal Diffusion

If the boundary of the domain,where a certain differential equation is solved,is unknown and needs to be determined as part of the solution,then such a problem is usually called a free boundary problem.It has been widely used in physics,chemistry,biology and medicine.The free boundary problem is not only more realistic than the fixed boundary problem,but its related theories are also important and difficult.Most classical dynamic systems involve the local effects of spatial structures(local diffusion or short-range effects).However,the nonlocal effects of the spatial structure(nonlocal diffusion or long-range effects)are common and very important.The existing research results show that for the same reaction term,if local diffusion and nonlocal diffusion are adopted respectively,the initial boundary value problems in the fixed domain have obviously different dynamic properties.This paper studies several types of free boundary problems with local or nonlocal diffusion,in order to find the differences between dynamics of free boundary problem with local and nonlocal diffusion.Chapter 1 consists of the research background and current situation,as well as the main contents of this thesis.In Chapter 2,we focus on a viral propagation model with a nonlinear infection rate and free boundaries.We assume that cells have no diffusion effect and virus particles adopt local diffusion strategy.Therefore,the model is made up of two ordinary differential equations and a free boundary problem of reaction-diffusion equation,and can also be regarded as the free boundary problem of reaction-diffusion equations with degenerate diffusion.Firstly,by the contraction mapping principle、the theory of ODEs and the theory of parabolic equations the existence and uniqueness of the global solution and the uniform estimate of the solution are given.Then the long-time behaviors of the solution are obtained by virtue of the basic theories of elliptic and parabolic equations and an iteration process.Finally,the criteria for spreading and vanishing are obtained by constructing the upper solution and using the comparison principle.Chapter 3 pays attention to the global solution and finite time blow-up of a free boundary problem with nonlocal diffusion.The existence and uniqueness of local solution is proved firstly.Then,with the aid of eigenfunction and comparison principle,the criteria for finite time blowup and the estimates of blowup rate and blowup time of blowup solution are obtained.At last,by constructing an upper solution and using the comparison principle,it is shown that the solution exists globally and decays exponentially as long as the initial function is small enough.In Chapter 4,we propose and study two nonlocal diffusion problems with one fixed boundary and one free boundary.What is different between the two models is the conditions imposed at the fixed boundary.More precisely,in the first model,we assume that the species can jump through the fixed boundary,but once they do it,they will die immediately.For the second model,similarly to the classical homogeneous Neumann boundary,the species is supposed not to cross the fixed boundary.When the nonlinear term is FisherKPP type,the existence and uniqueness of the global solution,the spreading-vanishing dichotomy together with the criteria for spreading and vanishing are given.When spreading occurs,for the first model,if the condition(J1)holds,the possible optimal estimation of spreading speed is obtained;for the second model,it is proved that the free boundary has finite spreading speed if and only if the condition(J1)is true.Chapter 5 continues to study the second model proposed in Chapter 4.By constructing suitable upper and lower solutions and using comparison principle,we first obtain more accurate longtime behaviors for solution than that in Chapter 4.Then it is proved that the limiting problem of the second model is a fixed boundary problem defined in halfspace R+ when expanding rate of free boundary converges to infinity.In addition,a sharp estimate on free boundary is derived for the kernel function with compact support set by employing some delicate upper and lower solutions,and for the kernel function which behaves like |x|-γ for γ∈(1,2]near infinity,we get sharp estimates on free boundary and the convergence of solution in large range.Chapter 6 investigates a mutualist model with nonlocal diffusion and free boundary.We first consider the model with a free boundary,and then extend the obtained results to the model with double free boundaries.The existence and uniqueness of global solution,spreading-vanishing dichotomy and criteria for spreading and vanishing are derived.Especially convergence for solution in large range and its detailed classification is obtained by constructing some delicate upper and lower solutions.