The reaction-diffusion equation is a very important class of mathematical models. In the study of reaction-diffusion equations, traveling wave solutions have been widely studied due to their importance in literature and application. Although many important results for traveling waves have been established, the intrinsic differences of models lead to rich dynamics, and the topic needs to be further studied in some systems. In this thesis, two types of diffusion models are considered from the viewpoint of their traveling wave solutions and the related propagation problems.
First, the initial value problems and the traveling wave solutions in an SIRS model with general generating functions are studied. Linearizing the infected equation at the disease free steady state, we can define a threshold if the corresponding basic reproduction ratio in kinetic system is larger than the unit. When the initial condition for the infected is compactly supported, we prove that the threshold is the spreading speed for three unknown functions. At the same time, this threshold is the minimal wave speed for traveling wave solutions modeling the disease spreading process. If the corresponding basic reproduction ratio in kinetic system is smaller than the unit, then we confirm the extinction of the infected and the nonexistence of nonconstant traveling waves. From the threshold, we may explore the factors that affect the spreading ability of the disease.
Second, a class of reaction diffusion competition models is then considered, which have degenerate diffusion and nonlinearity. The existence of a bistable traveling wave connecting two stable steady states is proved by the vanishing viscosity method. As a byproduct, an estimate of the wave speed is also given. Further, the comparison principle is used to prove the results of strict monotonicity and uniqueness of wave speed for traveling wave solutions. From the uniqueness, we may find some spreading properties in the corresponding initial value problems.