关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解

Formally we have arrived at the middle of the book. So you may need a pause for recovering, a pause which we want to fill up by some fixed point theorems supplementing those which you already met or which you will meet in later chapters. The first group of results centres around Banach’s fixed point theorem. The latter is certainly a nice result since it contains only one simple condition on the map F, since it is so easy to prove and since it nevertheless allows a variety of applications. Therefore it is not astonishing that many mathematicians have been attracted by the question to which extent the conditions on F and the space Ω can be changed so that one still gets the existence of a unique or of at least one fixed point. The number of results produced this way is still finite, but of a statistical magnitude, suggesting at a first glance that only a random sample can be covered by a chapter or even a book of the present size. Fortunately (or unfortunately?) most of the modifications have not found applications up to now, so that there is no reason to write a cookery book about conditions but to write at least a short outline of some ideas indicating that this field can be as interesting as other chapters. A systematic account of more recent ideas and examples in fixed point theory should however be written by one of the true experts. Strange as it is, such a book does not seem to exist though so many people are puzzling out so many results.

Fixed Point Theory

Diffusion and ecological problems : mathematical models

Models in Ecology

Many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. The mathematical description of these phenomena leads to impulsive differential equations. In this work we present a new approach via variational methods and critical point theory to obtain the existence of solutions to impulsive problems. We consider a linear Dirichlet problem and the solutions are found as critical points of a functional. We also study the nonlinear Dirichlet impulsive problem.

Variational approach to impulsive differential equations

https://www.math.miami.edu/%7Eruan/MyPapers/WangLiRuan-JDE06.pdf

Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays

This book summarizes the qualitative theory of differential equations with or without delays, collecting recent oscillation studies important to applications and further developments in mathematics, physics, engineering, and biology. The authors address oscillatory and nonoscillatory properties of first-order delay and neutral delay differential equations, second-order delay and ordinary differential equations, higher-order delay differential equations, and systems of nonlinear differential equations. The final chapter explores key aspects of the oscillation of dynamic equations on time scales-a new and innovative theory that accomodates differential and difference equations simultaneously.

Nonoscillation and Oscillation Theory for Functional Differential Equations

https://www.math.miami.edu/~ruan/MyPapers/WangLiRuan-JDE07.pdf

Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay

In this paper, we study the existence, uniqueness and stability of traveling wave fronts in the following nonlocal reaction–diffusion equation with delay

 $$\frac{\partial u\left(x, t\right)}{\partial t}= d\Delta u\left(x, t\right)+f\left(u\left(x, t\right),\int\limits_{-\infty }^\infty h\left(x - y\right) u\left(y, t - \tau\right) dy\right)\!.$$
 Under the monostable assumption, we show that there exists a minimal wave speed c* > 0, such that the equation has no traveling wave front for 0  c*, such a traveling wave front is unique up to translation and is globally asymptotically stable. When applied to some population models, these results cover, complement and/or improve a number of existing ones. In particular, our results show that (i) if ∂2
 f (0, 0) > 0, then the delay can slow the spreading speed of the wave fronts and the nonlocality can increase the spreading speed; and (ii) if ∂2
 f (0, 0) = 0, then the delay and nonlocality do not affect the spreading speed.

/pdf/traveling-fronts-in-monostable-equations-with-nonlocal-3j7taqnsl9.pdf

Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects

This paper is concerned with entire solutions for bistable reaction-diffusion equations with nonlocal delay in one-dimensional spatial domain. Here the entire solutions are defined in the whole space and for all time t ∈ R. Assuming that the equation has an increasing traveling wave solution with nonzero wave speed and using the comparison argument, we prove the existence of entire solutions which behave as two traveling wave solutions coming from both ends of the x-axis and annihilating at a finite time. Furthermore, we show that such an entire solution is unique up to space-time translations and is Liapunov stable. A key idea is to characterize the asymptotic behavior of the solutions as t → -∞ in terms of appropriate subsolutions and supersolutions. In order to illustrate our main results, two models of reaction-diffusion equations with nonlocal delay arising from mathematical biology are considered.

/pdf/entire-solutions-in-bistable-reaction-diffusion-equations-4j27aj2tht.pdf

Wan-Tong Li

Papers

Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays

Nonoscillation and Oscillation Theory for Functional Differential Equations

Existence and stability of traveling wave fronts in reaction advection diffusion equations with nonlocal delay

Traveling Fronts in Monostable Equations with Nonlocal Delayed Effects

Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity