Propagation Dynamics Of Evolution Equations In Heterogeneous Media

Many mathematical models for practical problems are evolution equations,such as the Schr?dinger equation in quantum mechanics,the Navier-Stokes equation in fluid mechanics,the reaction-diffusion equation in spatial ecology and lattice differential equation.As a bridge between the mathematical theory and the practical applications,it is of great practical significance to study evolution equations.An increasing attention has been paid to the propagation dynamics of heterogeneous evolution equations since the environment may be not homogeneous over time and/or space.The asymptotic speed of spread and traveling waves is one of the most important propagation dynamical issues in evolution equations.Although the space-time heterogeneity can better describe the complexity of the underlying environment,this also leads to the difficulty of applying the classical theory and concepts directly in heterogeneous media.To study the front propagation dynamics of evolution equations in heterogeneous media,one first needs to properly extend the notion of traveling wave solutions in the classical sense,so as to give the concept of generalized transition waves(semi-waves).It is of great importance to study the existence of generalized transition wave(semi-wave)solutions in heterogeneous media,and to understand the influence of the time and/or space heterogeneity on the wave profiles and wave speeds of such solutions.The topical question in the current thesis to gain the asymptotic speed of spread and the existence of generalized transition waves(semi-waves)for time and/or space heterogeneous models.Chapter 2 is concerned with the propagation dynamics of a nonlocal timespace periodic reaction-diffusion model with delay.We first prove the existence and global attractivity of time-space periodic solution for the model,and then by a family of principal eigenvalues associated with linear operators,we characterize the asymptotic speed of spread of the model in the monotone and non-monotone cases.Next,we introduce a notion of generalized transition semi-waves for the model,and then by constructing appropriate upper and lower solutions,and using the results of the asymptotic speed of spread,we show that there is a critical wave speed such that generalized transition semi-waves of the model in the non-monotone case exist when their wave speed is above a critical speed,and generalized transition semiwaves do not exist anymore when their wave speed is less than the critical speed.It turns out that the asymptotic speed of spread coincides with the critical wave speed of generalized transition semi-waves in the non-monotone case.In addition,the existence/non-existence of generalized transition waves in the monotone case can be obtained directly from the analysis in the non-monotone case.Finally,numerical simulations for various cases are carried out to support our theoretical results.In chapter 3,the propagation phenomena of a nonlocal delayed reaction-diffusion equation in time almost periodic media is considered.First,we study properties of exponentially decaying solutions for the corresponding linear problem and characterize the critical wave speed.Then,by constructing appropriate upper and lower solutions and using comparison arguments,we show that no matter the birth rate function is monotone or not,a generalized transition semi-wave exists as soon as the mean value of wave speed is greater than this critical speed.Meanwhile,some spreading properties for solutions with compact supported initial values are also established.Finally,a brief discussion is given to show that the critical speed of generalized transition semi-waves obtained in the present chapter coincides with the minimum speed of traveling waves observed by others for some special cases.In chapter 4,we study the existence of generalized transition waves and some properties of spreading speed intervals for a lattice differential equation with time and space dependence.Firstly,by constructing appropriate subsolutions and supersolutions and using comparison principal,we show that the existence of the critical speed of generalized transition wave for space periodic and time heterogeneous lattice Fisher-KPP equations,and prove that a generalized transition wave solution exists as soon as the least mean of wave speed is above this critical speed.And the critical speed we construct is proved to be minimal in some particular cases,such as space-time periodic media or time periodic and space homogeneous media.Finally,under the suitable assumptions,we give some properties of spreading speed intervals and the existence of generalized transition waves for the lattice differential equations in general time and space heterogeneous media.