摘要
在过去的几十年中,非线性偏微分方程理论的研究得到了极大的发展,而这些发展大多都是出于对生物学,物理学和化学等自然科学中的应用.本文主要讨论来自于生物学,物理学和化学中的几类发展方程的平衡态模式和解的渐近性质,包括解的周期性、爆破性及衰减性.全文分为三部分:
第一部分(本文第二章至第四章)主要讨论来自于生态学和化学中的几类反应扩散和分次扩散模型的平衡态模式.对于齐次Neumann边界条件,重点是研究一般扩散和分次扩散对模式生成(即非常数正平衡解)的影响;对于齐次Dirichlet边界条件,重点是分析正平衡解(共存解)的存在性、稳定性和渐近性.本部分的具体安排如下:
在第二章中,我们研究了two-cell Brusselator模型的模式生成问题.模式生成问题是现代科学技术中一个具有重要理论意义和实际应用背景的研究课题.它描述了自然界(如生态学、化学反应、基因生成)中几种物质相互作用时的结构变化.对于该模型,我们建立了正平衡解上下界的精确估计,得到了非常数正平衡解的存在性和不存在性条件.所得结果表明:在一定条件下,扩散能够导致模式. (本章的主要结果发表于J. Math. Anal. Appl., 2010 (366): 679–693)
在第三章中,我们考虑了一个带有齐次Dirichlet边界条件的捕食模型的平衡态模式.我们主要关心当参数变化时,共存解的存在性.众所周知,这些问题的研究是很有趣但通常是非常困难和具有挑战性的.通过细致分析解的渐近性态并借助拓扑度理论和分支理论,我们完全确定了该模型共存解存在的充要条件. (本章的主要结果发表于J. Math. Anal. Appl., 2010 (369): 555–563)
在第四章中,我们讨论了一个具有分次扩散的捕食模型的平衡态模式.我们首先对平衡态解作了一些先验估计,接着利用这些估计我们讨论了共存解不存在的条件,最后,利用分支理论,我们获得了共存解存在的充分条件并刻画了共存解的共存区域. (本章的主要结果已投往Math. Model Anal.)
第二部分(本文第五章至第七章)主要讨论非线性抛物方程解的周期性和爆破性.我们首先讨论的是周期问题.昼夜更替,日月变迁,生命繁衍,自然界许多状态或过程周而复始地有规律地变化着,因此对周期解的讨论有着重要的意义;其次,我们讨论非线性抛物方程解的爆破问题,对该问题的研究具有重要的理论意义和实际意义.这方面的研究是当今非线性发展方程理论研究中的前沿和热点问题之一.本部分的具体安排如下:
在第五章中,我们讨论了一个logistic型的多孔介质方程的周期解.利用Leray-Schauder不动点理论,我们首先建立了非平凡周期解的存在性,接着我们利用Moser迭代技术证明了这些周期解的支集和时间无关,最后利用单调性方法我们建立了极大周期解的吸引性. (本章的主要结果发表于Math. Meth. Appl. Sci., 2010 (33): 1942-1954)
在第六章中,我们讨论了齐次Dirichlet边界条件下的具有局部化源的弱耦合的退化、奇异抛物方程组解的爆破性质.我们首先证明了该方程组经典解的存在性,接着研究了解整体存在或有限时间爆破的充分条件,最后我们研究了爆破解的爆破集和爆破速率. (本章的主要结果发表于Z. Ange. Math. Phy., 2011 (62): 47-66)
在第七章中,我们研究了一类具有非局部源的非牛顿多方渗流方程解的整体存在性和爆破性.在一定的假设下,我们得到了解的整体存在或有限时间爆破的充分条件.最后,在临界条件下,我们同样讨论了该问题整体存在性和爆破性. (本章的主要结果发表于ANZIAM J., 2008 (50): 13-29)
第三部分(本文第八章至第九章)主要讨论非线性双曲方程解的爆破性、生命跨度和衰减性.由于双曲方程有着重要的物理背景,长期以来一直是数学工作者研究的热点.自从John引入生命跨度(局部解存在的最大时间)的概念以来,许多数学工作者在这方面做了大量的工作.他们主要讨论生命跨度与非线性项形式及空间维数之间的关系.本部分将讨论两类非线性双曲方程解的爆破性、生命跨度和衰减性.本部分的具体安排如下:
在第八章中,我们研究了一类含有非线性阻尼项的双曲系统的初边值问题.在一定的假设下,我们得到了全局解的不存在性并估计了解的生命跨度.我们的结果改进了Agre和Rammaha (Diff. Inte. Equation, 2006)的结果. (本章的主要结果已投往Nonlinear Anal.)
在第九章中,我们研究了一类具有阻尼项的非线性高阶波动方程.我们首先给出了整体解存在的充分条件,接下来,在一些初始能量的假设下,我们研究了局部解的爆破性质.对于整体解,我们研究了它的衰减性质,对于爆破解,我们研究了它的生命跨度.这些结果推广了Ye (J. Inequa. Appl., 2010)的结果. (本章的主要结果已投往Nonlinear Anal.)
Substantial progress had been made in the last two decades in the theory of nonlinear systems of partial differential equations. Much of the developments are motivated by applications to the natural sciences of biology, physics and chemistry. In this dissertation, we main discuss the stationary patterns, periodic, blowup and decay properties of some kinds of evolution equations which are came from biology, physics and chemistry. We divide the dissertation into three parts:
In the first part (Chapter 2-Chapter 4), we are concerned with the stationary patterns of some reaction-diffusion equations or fraction-diffusion equations. For the problem of Neumann boundary condition, we mainly study the effects of diffusion on the pattern formation (namely, positive non-constant steady-state solutions). When the boundary condition is of Dirichlet type, we mainly investigate the existence of coexistence solutions and stability of positive steady-state solutions (PSS), and determine the asymptotic behavior of PSS. The specific arrangements are as follows:
In Chapter 2, we consider the pattern formation of two-cell Brusselator model. Pattern formation now becomes an important research aspect of modern science and technology. It can be used to describe the structure changes of interacting species or reactants of ecology, chemical reaction and gene formation in nature. For the cited model, we establish the fine upper and lower bounds of PSS and then study the existence and non-existence of non-constant PSS. As a consequence, our results show that, under some cases, diffusion can create pattern formation. (The main results of this chapter are published in J. Math. Anal. Appl., 2010 (366): 679–693)
In Chapter 3, we consider the stationary patterns of a prey-predator model with Dirichlet boundary condition, and are mainly concerned about the existence of coexistence solutions. As it is known, such problems are very interesting in both mathematics and application, although they are usually quite difficult and full of challenges. By meticulously analyzing the asymptotic behaviors of solutions, we find the necessary and sufficient conditions to the existence of coexistence solutions by the classical Leray-Schauder degree theory and bifurcation theory. (The main results of this chapter are published in J. Math. Anal. Appl., 2010 (369): 555–563)
In Chapter 4, we discuss the stationary patterns of a Lotka-Volterra model with nonlinear diffusion of fraction type. First, we give some priori estimates for the steady state solutions. Second, we give some conditions for the non-existence of coexistence solutions by using the priori estimates. At last, we give some sufficient conditions for the existence of coexistence solutions by using the bifurcation theory. (The main results of this chapter are submitted to Math. Model Anal.)
In the second part (Chapter 5-Chapter 7), we discuss the periodic properties and blowup properties of nonlinear parabolic equations. We first discuss the periodic solutions, and then we study the blowup solutions. The specific arrangements are as follows:
In Chapter 5, we discuss the periodic solutions to a porous medium equation of logistic type. We establish the existence of nontrivial periodic solution by Leray- Schauder fixed point theory. We also show that the supports of these solutions are independent of time by providing a priori estimates for their upper bounds by using Moser iteration. Furthermore, we establish the attractivity of the maximal periodic solution by using the monotonicity method. (The main results of this chapter are published in Math. Meth. Appl. Sci., 2010 (33): 1942-1954)
In Chapter 6, we deal with the weakly coupled degenerate and singular parabolic equations with localized source. The existence of a unique classical non-negative solution is established and the sufficient conditions for the solution that exists globally or blows up in finite time are obtained. Furthermore, under certain conditions, we study the blowup set. At last, we also obtain the blowup rate under appropriate assumptions. (The main results of this chapter are published in Z. Ange. Math. Phy., 2011 (62): 47-66)
In Chapter 7, we deal with the global existence and blowup properties of the non-Newton polytropic filtration system with nonlocal source. Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depending on the initial data and the parameters; we also give a criterion for the solution exists globally or blows up in finite time in the critical case. (The main results of this chapter are published in ANZIAM J., 2008 (50): 13-29)
In the third part (Chapter 8-Chapter 9), we discuss the blowup, life-span and decay properties of hyperbolic equations. Since hyperbolic equations have strong physical background, many mathematic works focus on them for a long time. They study the relationship between the life-span and nonlinear terms or spatial dimensions since John introduced the concept of life-span (the existence time of local solution). In this part, we will discuss the the blowup, life-span and decay properties of two kinds of hyperbolic equations. The specific arrangements are as follows:
In Chapter 8, we study the initial-boundary value problem for a system of nonlinear hyperbolic equations, involving nonlinear damping terms, in a bounded domain ?. The nonexistence of global solutions is discussed under some conditions on the given parameters. Estimates on the lifespan of solutions are also given. Our results extend and generalize the recent results of Agre and Rammaha (Diff. Inte. Equation, 2006), specially, the blowup of weak solutions in the case of non-negative energy. (The main results of this chapter are submitted to Nonlinear Anal)
In Chapter 9, we consider a class of nonlinear higher-order wave equation with nonlinear damping in ?. We show that the solution is global in time under some conditions. We also show that the local solution blows up in finite time with some assumptions on the initial energy. The decay estimate of the energy function for the global solution and the lifespan for the blowup solution are given. These extend the recent results of Ye (J. Inequa. Appl., 2010). (The main results of this chapter are submitted to Nonlinear Anal)
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