High-Dimcnsional Traveling Fronts Of Bistable Reaction-Diffusion Equations

As a special type of parabolic equations, reaction-diffusion equation is a pow-erful tool for studying the nature of widespread diffusion phenomena. For example, we can use a reaction-diffusion equation to describe the heat conduction in physics, the combustion temperature problem in combustion theory, the material concen-tration change in chemical reaction, the species invasion in ecology and the spread of disease in the space etcetera. As a special type of solutions of reaction-diffusion equations, traveling wave solutions can well model the oscillatory phenomenon and the propagation with finite speed of nature. Therefore, the existence, uniqueness and stability of one-dimensional traveling wave solutions have been widely stud-ied. However, since many practical problems from physics, chemistry, ecology and other areas have occurred in the high-dimensional space, traveling wave solutions on high-dimensional space has been paid much more attention. In contrast to one-dimensional traveling wave solutions, the properties of high-dimensional traveling wave solutions become more complex, but also more meaningful. In fact, it has been found that there exist non-planar traveling wave solutions with a variety of different shapes level sets recently. So, the research of the high-dimensional trav-eling wave solutions and its qualitative properties is an interesting and challenging problem. This thesis is mainly concerned with the V-shaped traveling fronts of reaction-diffusion equations with nonlinear convection in two-dimensional space and non-planar traveling fronts of the Lotka-Volterra competition-diffusion sys-tem in high-dimensional space. The main contents of this thesis are as follows:Chapter 1 is an introduction to this thesis. We first summarize the background and the history of reaction-diffusion equations and its traveling wave solutions, and then we state the research problems and the main results of the present thesis.In Chapter 2, we studied the V-shaped traveling fronts of a reaction-diffusion equations with nonlinear convection in two-dimensional space. First of all, by using the supersolution and the subsolution, we construct a V-shaped traveling front, and show that it is the unique V-shaped traveling front between them. Then, we establish a series of estimates and construct different types of super-and sub-solutions to establish the global asymptotic stability of V-shaped traveling front.On the basis of Chapter 2, in Chapter 3, we further study the asymptotic stability and the instability of V-shaped traveling fronts in dimension n≥3. Our first result shows that the V-shaped traveling front is asymptotically stable under the initial perturbations decaying at infinity. In particular, for a class of initial value with special form, the V-shaped traveling front is algebraically stable, and the convergence rate is optimal in a certain sense. The second result shows that the V-shaped traveling front is also asymptotically stable, even if the initial perturbation dose not decay to zero at infinity but the initial value uo(x, y, z) satisfies some certain assumptions. Lastly, we use a counter-example to show that for the general initial perturbation which dose not decay to zero at infinity, the V-shaped traveling front may be unstable.In Chapter 4, we study the existence, nonexistence and the qualitative prop-erties of the conical traveling fronts of the Lotka-Volterra strong competition-diffusion system in R3. The basic assumption is that the wave speed c of the one-dimensional traveling front connecting two stable equilibria is positive. By constructing a sequence of pyramidal traveling fronts and then taking its limit, we obtain the conical traveling fronts. Finally, using the theory of spreading speed and the comparison principle we show a series of qualitative properties of the conical traveling fronts. Furthermore, we prove that there does not exist conical traveling front of the system with wave speed s< c, and then it shows that the curvature has an acceleration effect. The discussion in this chapter is a new method.Including Chapter 4, all of the results of high-dimensional traveling fronts of strong competition Lotka-Volterra diffusion system are applicable to the case that the one-dimensional wave speed c> 0. Therefore, in Chapter 5, we study the case that c= 0 and establish the existence, nonexistence and the qualitative of the V-shaped traveling fronts of the Lotka-Volterra strong competition-diffusion system in E2. Our method is to firstly construct a sequence of Lotka-Volterra competition systems with positive one-dimensional wave speeds and take a limit of conical traveling fronts of the systems, then prove the main results by using the arguments similar to those in Chapter 4.