【作者】 汪璇；

【导师】 钟承奎；

【作者基本信息】 兰州大学 ， 基础数学， 2009， 博士

【摘要】 在这篇论文中,通过运用无穷维动力系统关于吸引子理论的最新研究成果并且结合一些能量估计技巧,我们研究了两类方程:具有衰退记忆的非经典扩散方程和具有衰退记忆的半线性热方程.我们对其弱解和强解的长时间动力学行为进行了深入的讨论,并且证明了以上方程对应动力系统的全局吸引子或一致吸引子的存在性.首先,我们研究了在自治外力项作用下具有衰退记忆的非经典扩散方程u_t—△u_t—△u—∫₀^∞k（s）△u（t—s）ds=f（u）+g（x）.当非线性项f（u）满足临界指数增长,以及外力项g仅属于空间H^-1（Ω）或L²（Ω）时,通过应用半群分解技术和紧性转移定理来克服证明过程中存在的许多实质性困难,最终获得了方程的全局吸引子在弱拓扑空间和强拓扑空间的存在性结果（见定理3.2.9和定理3.3.8）.继而,我们研究了在非自治外力项作用下具有衰退记忆的非经典扩散方程u_t—△ut—△u—∫₀^∞k（s）△u（t—s）ds=f（u）+g（x,t）.当非线性项f（u）满足临界指数增长,以及符号空间仅为L_b²（R;L²（Ω））或L_b²（R;H₀¹（Ω））（而非平移紧）时,通过分解解过程,来进行解的渐近正则性估计,从而得到了方程在弱拓扑空间和强拓扑空间对应的过程族的紧的一致（w.r.t。g∈H（g₀））吸引子的存在性（见定理4.2.15和定理4.3.5）.最后,我们研究了在自治外力项作用下具有衰退记忆的半线性热方程u_t—△u—∫₀^∞k（s）△u（t—s）ds+f（u）=g（x）.在非线性项满足超临界指数增长条件下,通过应用抽象半群理论,进行先验估计,获得了解的存在唯一性,在此基础上利用收缩函数来验证解半群的渐近紧性,最终我们证明了方程的全局吸引子在弱拓扑空间L²（Ω）×L_μ²（R⁺;H₀¹（Ω））和强拓扑空间H₀¹（Ω）×L_μ²（R⁺;D（A））中的存在性（见定理5.2.9和定理5.3.7）.我们所研究的方程满足的条件很弱,关于非线性项通常是满足临界指数增长或超临界指数增长、而且在自治情形下外力项仅属于正则性较低的空间,在非自治情形下外力项仅为平移有界（符号空间不紧）,所以得到的结果极大地改进和推广了已有的一些结果.本文使用的主要工具为抽象半群理论,紧性转移定理及收缩函数.更多还原

【Abstract】 In this doctoral dissertation,applying the recent theoretical results about attractors and combining with some estimates of energy functional,we mainly consider the two classes of equations as follows:the non-classical diffusion equations with fading memory and the semilinear heat equations with fading memory.Long-time behavior of weak solution and strong solution is discussed and the existence of global attractors or uniform attractors respectively for either autonomous or non-autonomous case is gained.At first,we research the non-classical diffusion equations with fading memory u_t-△u_t-△u-integral from n=0 to∞k（s）△u（t-s）ds=f（u）+g（x） for the autonomous case.Because nonlinear term satisfies critical exponential growth condition and forcing term g only belongs to H^-1（Ω） or L²（Ω）,there exist some virtuality difficulties in the process of proof.Conquering above difficulties through decomposition technique of semigroup and compactness transition theorem, furthermore,we prove the existence of global attractors in weak topological space and in strong topological space（see Theorem 3.2.9 and Theorem 3.3.8）. After that,we discuss the non-classical diffusion equations with fading memory u_t-△u_t-△u-integral from n=0 to∞k（s）△u（t-s）ds= f（u）+g（x,t） for the non-autonomous case.When the nonlinear term satisfies critical exponential growth condition and the time-dependant forcing term is translation bounded（that is,only belongs to n_b²（R;L²（Ω）） or L_b²（R;H₀¹（Ω））） instead of translation compact,after decomposing the solution process we test and verify the asymptotic regularity of solutions.Based on the result we show the existence of compactly uniform attractors together their structure in both weak and strong topological spaces（see Theorem 4.2.15 and Theorem 4.3.5）.In the end,we consider the semilinear heat equations with fading memory u_t-△u-integral from n=0 to∞k（s）△u（t-s）ds+f（u）=g（x）.When the nonlinearity adheres to polynomial growth of arbitrary order,applying abstract semigroup theory,we make the priori estimate and obtain the existence and uniqueness of solutions. And then the asymptotic compactness of solution semigroup is gained by utilizing contract function theory.According to these results as indicated above, we show the existence of global attractors in both L²（Ω）×L_μ²（R⁺;H₀¹（Ω）） and H₀¹（Ω）×L_μ²（R⁺;D（A））（see Theorem 5.2.9 and Theorem 5.3.7）.Since the equations we study satisfy the weaker assumptions as follows:the nonlinearity adheres to critical exponential growth or polynomial growth of arbitrary order;and for the autonomous case,the forcing term only belong the lower regularity space,or for the non-autonomous case,the forcing term is only translation bounded,these results we gain improve and extend some known results extremely.The principal tools in my paper are:abstract semigroup theory,compactness transition theorem and contract function.更多还原