| A lot of mathematical models derived from various fields, such as biology, physics,chemistry and ecology, are the so called reaction-diffusion equations. In recent years,the theory of delayed reaction-diffusion equations have been widely studied. Traveling wave solutions is a special type of solutions of the reaction-diffusion equations. It can not only depict many properties of the equations but also describe various phenomena such as the disease spread and species invasion. Therefore, it is very important to investigate the traveling wave solutions of the reaction-diffusion equations with time delays. The thesis mainly studied the existence, uniqueness and stability of the traveling wave solutions of two types of monostable reaction-diffusion equations with time delays.Firstly, we study a type of monostable reaction-diffusion equations with time delays. In the case where the reaction term does not satisfy the sub-tangent condition, by constructing appropriate sub- and super-solutions and applying the squeezing technique,the global asymptotic stability of the minimal traveling wave solutions(i.e., the traveling wave solutions with a minimal wave spread) is proved.Secondly, we study a type of monostable cellular neural network model(space discrete delayed reaction-diffusion equations). We first determine the exact asymptotic behavior of the traveling wave solutions at infinity using Ikehara’s theorem. As a by-product, we obtain the non-existence of the traveling wave solutions. Then, based on a strong comparison principle, we proved the monotonicity and the uniqueness(up to translation)of the traveling wave solutions. |
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