Traveling Waves And Entire Solutions For Nonlocal Dispersal Equations

The theory of reaction diffusion equations is an important component of mod-ern mathematics researches. And the diffusion terms of classical reaction diffusion equations are described by Laplace operators. However, Laplace operators only re-flect local interactions in space. Actually, the effect of nonlocal dispersal in space is universal in nature. For example, in a biological population, species move in a wide expanse of space not only in a small one, which leads to the effect of nonlocal dispersal. Recently, in ecosystem, epidemiology, materials sciences and so on, a large number of diffusion equations with nonlocal effect which is embodied by convolu-tion operator have been established. Such equations are called nonlocal dispersal equations. However, the effect of nonlocal dispersal not only leads to mathematical difficulties, but also essential changes of dynamics. For example, the effect of nonlo-cal dispersal may make the maximum principle fail, make the regularity of solutions lower, make minimal wave speed of traveling wave solutions increase, may make traveling wave solutions be not unique(up to translation), and may cause leaping traveling wave solutions, etc. Therefore, it is more meaningful and valuable in theory and practice to study such equations. In this thesis, we consider traveling wave solu-tions and entire solutions of nonlocal dispersal equations in one-dimensional space. Here, the entire solutions are defined in the whole space and for all time t∈(?).Our thesis firstly consider the traveling wave solutions of nonlocal anisotropic dispersal equations with monostable nonlinearity. Usually, while considering the traveling wave solutions of nonlocal dispersal equations, people first prove the ex-istence of traveling wave solutions, then study their exponential decay at infinity. Thus, fairly good properties of the kernel functions of the convolution operators are required. In order to weaken requirements of kernel functions, we directly consider the existence of traveling wave solutions with exponential decay. At first, convert the existence problem of traveling wave solutions of the equation into a fixed point problem of a suitable operator, then the existence of traveling wave solutions which decay exponentially at infinity is obtained via sub-super solutions method coupled with the monotone iterative technique by constructing suitable sub-super solutions. In addition, we prove the uniqueness of traveling wave solutions by using a mov-ing plane technique. Furthermore, to prove the globally asymptotically stability of traveling wave solutions, we establish a series of properties of solutions of cauchy problem and construct proper sub-super solutions.Next, we consider the entire solutions of nonlocal dispersal equations. Because nonlocal dispersal operator make the regularity of solutions lower, in order to obtain a convergent subsequence of solution sequence of Cauchy problem, we make higher demands on the nonlinearities, which make the solutions are Lipschitz with space variable. Then the the existence of entire solutions with five parameters, four param-eters and three parameters of nonlocal isotropic dispersal equations with monostable nonlinearity is established by combing two traveling wave solutions with different speeds and coming from both ends of the real axis and some spatially independent solutions. We also gain their some properties.Finally, we study entire solutions of nonlocal isotropic dispersal equations with bistable nonlinearity. In order to construct entire solutions, we first get a priori decay rate at both ends of real axis of traveling wave solutions. Then a two-dimensional manifold of entire solutions is established by constructing proper sub-super solutions and using comparison principle. Furthermore, uniqueness and Liapunov stable are also showed.