Propagation Phenomena Of Diffusive KPP Equations With Free Boundaries In Time Almost Periodic Environments

This dissertation is concerned with propagation phenomena of diffusive KPP equa-tions with free boundaries in time almost periodic environments. More specifically, we first study the asymptotic dynamics of diffusive KPP equations in time almost periodic environments, and then prove the spreading-vanishing dichotomy for such equations with free boundaries. Furthermore, in the special case of spatially homogeneous en-vironments, we show that when spreading occurs, the spreading speeds are uniquely determined by the time almost periodic semi-wave solutions.The main results of the dissertation consist of the following five parts.In Chapter 1, we briefly introduce reaction-diffusion equations and its applications in ecology. And then recall the recent development of the free boundary problems. We finalize the chapter by listing the main results of the dissertation.In Chapter 2, we recall the definitions and properties of almost periodic functions, principal Lyapunov exponent and part metric associated to diffusive KPP equations. Moreover, for free boundary problems, we recall the comparison principle and zero number properties.In Chapter 3, we study the asymptotic dynamics of diffusive KPP equations in general spatially heterogeneous and time almost periodic environments. We first re-call the existence, uniqueness and stability of time almost periodic positive solution of diffusive KPP equations in fixed bounded domain, assuming that the linearized equa-tions have principal Lyapunov exponents. Next, by applying the theories mentioned above, we show the existence and stability of an almost periodic positive solution in fixed unbounded domain.In Chapter 4, we prove that a spreading-vanishing dichotomy holds for diffusive KPP equations with free boundaries in spatially heterogeneous and time almost periodic environments. More precisely, by choosing the spreading front and spreading ability as the parameters, we obtain some sufficient conditions for spreading and vanishing.In Chapter 5, for spatially homogeneous and time almost periodic environments, an more important result is proved, that is, the existence, uniqueness and stability of time almost periodic positive semi-wave solutions. And then we show that the spreading speeds of the free boundaries problems coincide with the speeds of the corresponding semi-waves. More specifically, based on the zero number arguments and part metric properties, we first prove some basic properties of diffusive KPP equations in unbound-ed domains with free boundaries and in fixed unbounded domains, the relationship be- tween these two equations is also discussed. Secondly, we prove the existence, unique-ness and stability of time almost periodic positive semi-wave solutions. Furthermore, we make use of semi-wave solutions to determine the spreading speeds. Finally, we deal with the double fronts case by simply modifying the techniques developed in the single front case.