| Diffusion in heterogeneous spatiotemporal environment has a profound application background in ecology and other fields,and is a research hotspot of nonlinear diffusion equation.Unfavorable environment is a kind of characteristic and meaningful heterogeneous space environment.This paper studies the qualitative properties of the solutions of reaction-diffusion equation in one-dimensional space with unfavorable environment,including:reaction-diffusion equation in the environment with fixed unfavorable or directional mobility unfavorable,or the free boundary of reaction-diffusion equation encounter unfavorable environment,etc,and studies the influence of the corresponding unfavorable environment on the spreading of solutions.This paper is divided into four chapters.In Chapter 2,we study a model in which the initial value like traveling wave moving to the right encounters a fixed unfavorable environment.We will prove that there is a critical width of unfavorable environment:spreading always occurs when the width of unfavorable environment is less than this critical value;When the width of unfavorable environment is equal to this critical value,the asymptotic behavior of the solution has a Spreading-Blocking dichotomy result;When the width of unfavorable environment exceeds this critical value,the Spreading-TransitionBlocking trichotomy results are valid.Furthermore,we also characterize the profile and speed of the solution at which spreading occurs.In Chapter 3,we study the unfavorable environment model of directional movement,focusing on the influence of the unfavorable environment moving to the right at a fixed speed on the spreading phenomenon.We will prove that there is a critical speed of unfavorable movement:when the speed of unfavorable environment movement to the right is less than this critical speed,the influence of unfavorable environment on solution spreading is similar to that of fixed unfavorable environment;When the speed of the unfavorable environment moving to the right is equal to this critical value,the solution of the problem will move to the right after the unfavorable environment and produce a certain lag;When the speed of the unfavorable environment moving to the right exceeds this critical value,the unfavorable environment has no effect on the spreading of the solution(that is,the unfavorable environment is far ahead and the solution spreading freely after the unfavorable environment).The same can be said for the case where the unfavorable environment moves to the left.But more interesting things can be found.For example,if the unfavorable environment is moving to the left at a critical speed,the trichotomy results will be true.This is fundamentally different from the case where the unfavorable environment moves to the right at critical speed.In Chapter 4,we consider the model of the free boundary encountering a fixed unfavorable environment,that is,whether the solution of the reaction-diffusion equation with a free boundary can rush through the barrier and spread.We show that the any solutions of the model can only converge to one of four different types of equilibrium solutions,and we give sufficient conditions for whether the free boundary can or cannot successfully cross the unfavorable environment. |
