Study On Some Nonlinear Problems Related To Free Boundary

The free boundary problem is a special qualitative issue of partial differential equations. This kind of problem mostly come from the real world or the applications of other science, for example:tumor growth problem, the American option prizing problem, melting solidification problem of the metallurgy industry, dam seepage problem, crystal growth problem, image processing, wound healing, the spreading of populations, and so on. Investigations of free boundary problem, add great help in explaining many nonlinear phenomena:Meanwhile, it promotes the development of mathematical theory and methods, and thus can serve as powerful tools for exploring such kind of equations that appear in the scientific technology areas. Thus it becomes an important addition to the fast development and application of nonlinear partial differential equations theory. Therefore, free boundary problems have attracted much attention in the field of partial differential equations.Another kind of problem describing the competition of populations in ecological systems can also derive a free boundary problem under some conditions, although they are operating on fixed spatial domains. The dynamic system models, describ-ing the co-existence or competition of populations by means of ordinary or partial differential equations, have been widely discussed. Recently, many mathematics researchers are interested in spatial segregation between two competing species.The present dissertation focuses on the region that changes as time proceeds and studies three free boundary problems, which are discussed in five chapters.Chapter1introduces briefly the background and development about the rele-vant work.Chapter2deals with two cases of a superlinear heat equation with free bound-ary. The first case looks at the multi-dimensional free boundary problem. The local existence and uniqueness of classical solution are proved by means of contraction mapping theorem. It is achieved by transforming the free boundary problem to a parabolic problem on a fixed domain. Then, we study the asymptotic behavior and define an energy function that is correlated with the initial values and space dimension. Then based on the establishment of several energy identities, an energy condition which is related with initial value and space dimension is derived. We prove that, under this condition, the solution of the problem blows up in a finite time. Finally, by investigating the properties of global solutions, we give the classi-fication of global solution. Results show that, if initial value is small, the solution exists and is global. The free boundary is asymptotically bounded, and the solution is called the global fast solution; if initial value is appropriate, there exists a global slow solution, namely, free boundary goes to infinite as time tends to infinite. In the second case, we explore the dynamical behavior of the solutions to a heat equation with double fronts free boundary in one-dimensional case. We first set up some preliminary lemma, including local existence and uniqueness of the classical solu-tion, monotonicity properties of the two free boundary; and their special property:2h0<g(t)+h(t)<2h0, t∈[0, t*), where the two free boundary fronts g(t) and h(t) are both finite or infinite at the same time. Then we show that blowup occurs if the initial datum is large enough. Moreover, all global solution is uniformly bounded and decays uniformly to0. The sufficient conditions for the existence of global fast solution and slow solution are given.Chapter3is contributed to an epidemic model describing the transmission of diseases with free boundary. I first, present some results derived from ordinary and partial differential equation on a fixed domain, and carry out numerical simulations. Then, the behavior of positive solutions to a reaction-diffusion system in a radially symmetric domain is studied. The existence and uniqueness of the global solution are given by the contraction mapping theorem. In addition, sufficient conditions for the disease vanishing or spreading are derived. Results show that the disease will not spread to the whole area if the basic reproduction number Ro<1or the initial infected radius h0is sufficiently small even that R0>1. However, the disease will spread to the whole area if if R0>1and h0is suitably large. This result is different from that obtained from ordinary or partial differential equations with fixed domains.Chapter4discusses a Volterra-Lotka competition model of quasilinear parabolic equations with large interaction. The model can derive a limiting problem which turns out to be a free boundary problem. In ecological systems, because of the influence of the cyclical environment, some of the biological habits also show the periodicity, which affects the number of population, and cyclical phenomena may appear. Many scholars have focused on the existence and stability of periodic solu-tion of these models. In the first part chapter, the competition model of quasilinear parabolic system with periodic coefficients is first investigated. Through the method of upper and lower solutions and its associated monotone iterations, the existence and asymptotic behavior of periodic solution are proved. Also, we show that the problem admits a maximal periodic solution and a minimal periodic solution. Then the sufficient conditions of the existence of periodic solution are derived. In the sec-ond part, we first set up the interior estimates together with the uniform estimates for the positive solution to a quasilinear parabolic system with large interaction. Then the two competing species exhibit spatial segregation in the case that the interspecific competition rates are sufficiently large. That is, the density of up to one population is not equal to zero, another type of population must be extinct. More importantly, the solution{uk},{vk} converges to the solution of a free bound-ary problem, as kâ†'∞. Finally, we prove that the subsequences{uk} and{vk} of k-dependent non-negative solutions converge strongly in L2(0, T; H1(Ω)) when self-diffusion coefficient α11=α22=0.The last chapter summarizes the above results and presents further considera-tions.