Free Boundary Problems Of Several Reaction-diffusion Systems

Many natural phenomena,such as,heat conduction in physics,change of substance concentration in chemical reaction and expansion of a new or inva-sive species in ecology,could lead to reaction-diffusion equation,which is a typ-ical class of semi-linear parabolic partial differential equations.Free boundary means that the region of partial differential equations is unknown and given with the solution together.The studies of reaction-diffusion system with free boundary are one of the important aspect of reaction-diffusion system researches.In this thesis,we consider free boundary problems of reaction-diffusion systems,and give spreading-vanishing dichotomy,spreading?virtual spreading?-transition-vanishing trichotomy,some sufficient conditions for spreading and vanishing,an estimate for the asymptotic spreading speed of the free boundary when spreading happens,and apply free boundary problems to describe information diffusion in social networks and dengue transmission.Firstly,a reaction-advection-diffusion equation with Robin and free bound-ary conditions is investigated.Establishing the comparison principle,we obtain the long time behavior of the solution by constructing suitable upper and lower solutions.We present a spreading-vanishing dichotomy and obtain criteria for spreading and vanishing.When spreading happens,we make use of the semi-wave method to derive the asymptotic spreading speed of the free boundary when spreading occurs.Numerical simulation is given to illustrate the impacts of the initial area and the expansion capacity on the free boundary.Then,a reaction-advection-diffusion equation is studied with double free boundaries and mth-order Fisher nonlinearity.The main purpose is to study the influence of the advection on the long time behavior of this problem.We obtain a rather complete description,that is,a spreading-transition-vanishing trichotomy with small advection,a virtual spreading-transition-vanishing trichotomy with medium-sized advection,and vanishing happens with large advection.When spreading happens,we prove that the leftward and rightward asymptotic spread-ing speeds are strictly decreasing with respect to m and display the uniform convergence of the spreading solution.Numerical simulations show the impacts of the advection and the initial value on the free boundaries.Next,a reaction-diffusion system with free boundary is proposed to illustrate multiple information diffusion in online social networks.We prove a spreading-vanishing dichotomy for the information:i.e.,information either last forever or suspend in finite time.The criterion for spreading and vanishing is obtained,which depends on the initial value and the expansion capacity.We present four cases of the long time behavior of the information according to the interaction between information.When the information spreads,we provide some upper and lower bounds of the asymptotic spreading speed corresponding to different cases of the long time behavior of the information.Numerical examples are given to illustrate the impacts of the initial area and the expansion capacity on the free boundary,and all cases of the long time behavior of the information.We investigate a free boundary problem to describe the transmission of the dengue at last.In addition to the classical basic reproduction number R₀,we introduce a new basic reproduction number R₀^F?t?related to time and present its properties.R₀^F?t?is associated with the principle eigenvalue of a correspond-ing eigenvalue problem.In view of the comparison principle and R₀^F?t?,some sufficient conditions for the disease vanishing and spreading are obtained by con-structing suitable upper and lower solutions.Numerical simulations illustrate the theoretical results and present an estimate for the asymptotic spreading speed of the free boundary.