兰州大学机构库 >数学与统计学院
非局部扩散Logistic方程的正解
Alternative TitleOn the Positive Solutions of Non-local Dispersal Logistic Equations
孙建文
Thesis Advisor李万同教授
2013-11-29
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name博士
Keyword非局部扩散方程 上下解方法 稳态解 正周期解
Abstract首先研究非局部扩散算子的主特征值, 对于线性非局部扩散算子,给出了主特征值存在的充分必要条件及带参数特征值方程的单调极限. 其次, 研究了空间退化的非局部扩散Logistic 方程正稳态解.通过证明辅助单调序列的极限问题, 得到退化Logistic 方程的正上解. 接着由非局部估计证明了正稳态解的唯一性.最后利用上下解方法结合比较原理得到了正稳态解的渐近稳定性以及发展方程解的长时间行为.接着我们研究一类非局部扩散Selection-Migration 基因模型.为此首先证明了稳态解稳定性的判别条件, 从而得到非平凡稳态解存在的必要条件以及非平凡稳态解的唯一性. 然后利用稳定性方法结合辅助方程证明了非平凡稳态解的存在及稳定性. 最后, 研究了时间周期的非局部扩散Logistic 方程的正周期解. 通过证明辅助特征值问题主特征值的单调极限得到正周期解的存在性. 然后利用比较讨论得到正周期解的唯一性. 同时, 通过上下解方法得到初值问题解的长时间行为及扰动周期非局部扩散方程解的渐近行为.特别地, 我们发现时间退化不会引起周期非局部扩散Logistic 方程动力学行为的变化; 然而空间退化与时空退化使得动力学行为发生变化. 进一步, 我们还发现空间退化与时空退化对发展问题产生了同样的影响.
Other AbstractFirstly, we study the principal eigenvalues of linear non-local dispersal operator and time periodic non-local dispersal operator. Since there is a deficiency of compactness of non-local operators as well as its semiflows of evolution problems, we know that the non-local dispersal operators may not admit a principal eigenvalue. We establish the conditions on the existence of principal eigenvalues of linear non-local operators and consider the corresponding sequences limit. Meanwhile, we find that there always exists a principal eigenvalue if there presents a spatial degeneracy. We give the monotonicity and sub bound of the periodic principal eigenvalues, which provide an effective tool for the study of positive solutions of non-local Logistic equations. In the second part, we consider the existence, uniqueness and asymptotical stability of positive steady states of the spatially degenerate non-local dispersal Logistic equations. After establishing a limit problem of auxiliary monotone sequences, we obtain the existence of positive steady states. The uniqueness is obtained by the non-local estimates. Then we use the sub-super solutions method and comparison principle to show the asymptotical stability of positive steady states and long time behaviors of the evolution equations. It is also shown that the spatial degeneracy makes a fundamental change of the dynamics behaviors of non-local dispersal Logistic equations. Thirdly, we consider a non-local dispersal equation arising from a selection-migration model in genetics, which is a non-local Logistic type equation with a weight function. Similarly to the classical selection-migration equation, it is difficult to construct a nontrivial super solution. We change mind to give a criterion on the stability of steady states. This change of mind is effective to show the uniqueness of nontrivial steady state. It also provides a necessary condition on the existence of nontrivial steady state. The existence is proved by the auxiliary equations combined with stability method. Finally, this thesis deals with the positive periodic solutions of time-periodic non-local dispersal Logistic equations. The periodic sub-super solutions method is proved by the monotone iterations of the corresponding initial value problems. If there has a temporal degeneracy, we obtain the existence of positive periodic solutions by the sub bound estimate of principal eigenvalues. If there presents spatial degeneracy or spatiotemporal d...
URL查看原文
Language中文
Document Type学位论文
Identifierhttps://ir.lzu.edu.cn/handle/262010/225473
Collection数学与统计学院
Recommended Citation
GB/T 7714
孙建文. 非局部扩散Logistic方程的正解[D]. 兰州. 兰州大学,2013.
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