连续与离散反应扩散方程组的行波解及整体吸引子
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摘要
物理、化学和生物等领域中的许多模型都可归结为反应扩散方程组的初值问题或初边值问题。格微分方程(离散的反应扩散方程)在材料科学、图像处理、化学反应、生物学等学科领域中都有应用。行波解是连续的反应扩散方程和格微分方程的一类很重要的解。偏微分方程空间变量离散后解的形态研究近期已引起人们极大的兴趣，对相应离散模型的研究有助于数值计算和数值分析，并可以得到无穷维动力系统和相应的有限维离散模型的密切联系等等。本文研究了时滞反应扩散方程、时滞格微分方程(组)、扩散Predator-Prey模型等系统行波解的存在性和FitzHugh-Nagumo方程在不同边值条件下空间离散或时空离散后解的渐近行为。
对于时滞反应扩散方程，我们先利用吴建宏和邹幸福[J. Dynam. Diff. Eqns 2001(3)]中的主要定理来研究时滞竞争扩散Lotka-Volterra系统波前解的存在性，给出了这个定理在非线性项满足弱拟单调条件(QM*)时在系统情况中的应用；并利用单调迭代方法和上、下解技术，对于具有部分零扩散系数的时滞反应扩散方程建立波前解的存在性定理，对于具有部分零扩散系数的时滞反应扩散方程建立波前解的存在性定理。从而使吴建宏和邹幸福[J. Dynam. Diff. Eqns, 2001(3)]的理论进一步完善。
对于具有“弱拟单调”非线性项的时滞反应扩散系统(包括正扩散系数和部分零扩散系数两种情况)，我们利用Schauder不动点定理，在适当构造的、赋予指数衰减范数的某个子空间上证明了波前解的存在性，使得上解不必一定是单调的，不必一定满足左极限条件，且上、下解的部分分量可取成常数，从而在构造上、下解时要容易些，并将所得的结论应用到带两个时滞的Belousov-Zhabotinskii模型上。
对于部分解耦的时滞反应扩散方程(吴建宏和邹幸福[J.Dynam. Diff. Eqns, 2001(3)]的理论不能应用到该类模型上)，我们根据其非线性项对变元的不同单调性，利用不同的上、下解的定义，引入交叉迭代格式，利用Schauder不动点定理证明其行波解的存在性，解决了一类时滞反应扩散方程组的行波解的存在性问题。
对于时滞格微分方程，利用Schauder不动点定理证明了其波前解的存在性，减少了对上解的限制条件，给出了上、下解在有限个点不连续时，行波解存在的条件；并将时滞反应扩散方程的结论推广到相应的时滞格微分方程组上。
对于反应项是Holling-Ⅱ型的扩散Predator-Prey模型，我们在月4中利用打
F＼博士学住论文
④DOCTOICh．－－－ DISS二RTATION
靶法，结合Lasalle不变原理和Liop。m。函数的万法，证明了扩散P陀da切，一P陀y
模型的行波解的存在性及小振幅行波链解的存在性。
对于FitzH。lgl。析叱unl。万程，将其在Dirichl毗和N刘m川ll边值条件下空间或
时空离散后，采用扰动的办法而非将相空间分解两个正交子空问的方法，证
明了整体吸引子的存在性，给出了其与分割点无关的Hausdorff维数的上界估
计；并将二个神经元细胞耦合 FitzHugll朴agumo模型推广到 n…三 2）神经元细
胞耦合RtzH。吧l朴旭u；n。模型，并讨论了广义耦合＊tm屹朴旭um。万程解的渐
近行为，给出了其整体吸引子的存在性和Ha朋d扯咐维数估计；最后对离散耦
合 FitzHugll－N。g。。。。。口万程也作类似白讨论。
This dissertation investigates both existence of traveling wave solutions for delayed reaction diffusion systems and lattice differential equations, and global attractor of spatially discretized FitzHugh-Nagumo equations with Dirichlet or Neumann boundary conditions.
For delayed reaction diffusion systems, the existence of traveling wavefronts in diffusive and coorperative system with time delays is provided, firstly; the monotone iteration scheme, together with upper-lower solution technique, is applied to establish the existence of traveling wavefronts of delayed reaction diffusion systems with some zero diffusive coefficients. Secondly, Schauder fixed point theorem is applied to some operators to prove the existence of traveling wave solutions in a properly subset equipped with exponential decay norm, which is obtained from a pair of upper and lower solutions for delayed reaction diffusion systems with non-quasimonotoiiicity. Both positive diffusive coefficients and some zero diffusive coefficient cases are considered. Finally, for partial decoupling systems, such as competitive-cooperative model with time delay and delayed epidemic model, we employ the new cross-iteration method, together with Schauder fixed point theorem, to establish the existence of traveling wave solutions. Both partial quasi-monotonicity and partial non-quasimonotonicity cases are considered separately. Similar results are obtained for some delayed lattice differential equations and systems of delayed lattice differential equations.
For diffusive Predator-Prey model without time delay, shooting argument is applied, together with unstable manifold theorem and LaSalle's Invariance Principle to prove the existence of traveling wave solutions in R4.
For spatially discretized FitzHugh-Nagumo equations with Dirichlet boundary condition or Neumann boundary condition, the existence of global attractor and its Hausdorff dimensions are proved separately. The result shows that the upper bounded estimate is independent of the number of spatially partition. Similar results are obtained for spatial-temporal discretized FitzHugh-Nagumo equations and generalized coupled FitzHugh-Nagumo equations.
对于时滞反应扩散方程，我们先利用吴建宏和邹幸福[J. Dynam. Diff. Eqns 2001(3)]中的主要定理来研究时滞竞争扩散Lotka-Volterra系统波前解的存在性，给出了这个定理在非线性项满足弱拟单调条件(QM*)时在系统情况中的应用；并利用单调迭代方法和上、下解技术，对于具有部分零扩散系数的时滞反应扩散方程建立波前解的存在性定理，对于具有部分零扩散系数的时滞反应扩散方程建立波前解的存在性定理。从而使吴建宏和邹幸福[J. Dynam. Diff. Eqns, 2001(3)]的理论进一步完善。
对于具有“弱拟单调”非线性项的时滞反应扩散系统(包括正扩散系数和部分零扩散系数两种情况)，我们利用Schauder不动点定理，在适当构造的、赋予指数衰减范数的某个子空间上证明了波前解的存在性，使得上解不必一定是单调的，不必一定满足左极限条件，且上、下解的部分分量可取成常数，从而在构造上、下解时要容易些，并将所得的结论应用到带两个时滞的Belousov-Zhabotinskii模型上。
对于部分解耦的时滞反应扩散方程(吴建宏和邹幸福[J.Dynam. Diff. Eqns, 2001(3)]的理论不能应用到该类模型上)，我们根据其非线性项对变元的不同单调性，利用不同的上、下解的定义，引入交叉迭代格式，利用Schauder不动点定理证明其行波解的存在性，解决了一类时滞反应扩散方程组的行波解的存在性问题。
对于时滞格微分方程，利用Schauder不动点定理证明了其波前解的存在性，减少了对上解的限制条件，给出了上、下解在有限个点不连续时，行波解存在的条件；并将时滞反应扩散方程的结论推广到相应的时滞格微分方程组上。
对于反应项是Holling-Ⅱ型的扩散Predator-Prey模型，我们在月4中利用打
F＼博士学住论文
④DOCTOICh．－－－ DISS二RTATION
靶法，结合Lasalle不变原理和Liop。m。函数的万法，证明了扩散P陀da切，一P陀y
模型的行波解的存在性及小振幅行波链解的存在性。
对于FitzH。lgl。析叱unl。万程，将其在Dirichl毗和N刘m川ll边值条件下空间或
时空离散后，采用扰动的办法而非将相空间分解两个正交子空问的方法，证
明了整体吸引子的存在性，给出了其与分割点无关的Hausdorff维数的上界估
计；并将二个神经元细胞耦合 FitzHugll朴agumo模型推广到 n…三 2）神经元细
胞耦合RtzH。吧l朴旭u；n。模型，并讨论了广义耦合＊tm屹朴旭um。万程解的渐
近行为，给出了其整体吸引子的存在性和Ha朋d扯咐维数估计；最后对离散耦
合 FitzHugll－N。g。。。。。口万程也作类似白讨论。
This dissertation investigates both existence of traveling wave solutions for delayed reaction diffusion systems and lattice differential equations, and global attractor of spatially discretized FitzHugh-Nagumo equations with Dirichlet or Neumann boundary conditions.
For delayed reaction diffusion systems, the existence of traveling wavefronts in diffusive and coorperative system with time delays is provided, firstly; the monotone iteration scheme, together with upper-lower solution technique, is applied to establish the existence of traveling wavefronts of delayed reaction diffusion systems with some zero diffusive coefficients. Secondly, Schauder fixed point theorem is applied to some operators to prove the existence of traveling wave solutions in a properly subset equipped with exponential decay norm, which is obtained from a pair of upper and lower solutions for delayed reaction diffusion systems with non-quasimonotoiiicity. Both positive diffusive coefficients and some zero diffusive coefficient cases are considered. Finally, for partial decoupling systems, such as competitive-cooperative model with time delay and delayed epidemic model, we employ the new cross-iteration method, together with Schauder fixed point theorem, to establish the existence of traveling wave solutions. Both partial quasi-monotonicity and partial non-quasimonotonicity cases are considered separately. Similar results are obtained for some delayed lattice differential equations and systems of delayed lattice differential equations.
For diffusive Predator-Prey model without time delay, shooting argument is applied, together with unstable manifold theorem and LaSalle's Invariance Principle to prove the existence of traveling wave solutions in R4.
For spatially discretized FitzHugh-Nagumo equations with Dirichlet boundary condition or Neumann boundary condition, the existence of global attractor and its Hausdorff dimensions are proved separately. The result shows that the upper bounded estimate is independent of the number of spatially partition. Similar results are obtained for spatial-temporal discretized FitzHugh-Nagumo equations and generalized coupled FitzHugh-Nagumo equations.
引文
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