一类四室非线性系统的整体存在性和长时间动力学
Global Existence and Long time dynamics of a four compartment Brusselator Type system
摘要：
在这项工作中，我们考虑了一个四室的选择器系统。由于该系统的反应项是变号的，因此解的分量不是先验有界的，用泛函方法推导先验界是不可行的。通过构造适当的李雅普诺夫泛函，我们证明了解的整体存在性。此外，由于符号变化的非线性，渐近符号条件也不满足，造成进一步的困难，证明全局吸引子的存在性。这些困难是规避通过使用构造的李雅普诺夫函数以及使用的一致格隆沃尔引理。我们证明了该系统吸引子的存在性，改进了文献中已有的结果，并证明了吸引子的Hausdorff维数和分形维数是有限的，特别地，我们得到了全局吸引子的豪斯多夫维数的一个下界.我们使用数值模拟，以及通过非线性时间序列分析的数值吸引子重建方法，以验证我们的结果。
英文摘要：
In this work we consider a four compartment Brusselator system. The reaction terms of this system are of non constant sign, thus components of the solution are not bounded apriori, and functional means to derive apriori bounds will fail. We prove global existence of solutions, via construction of an appropriate lyapunov functional. Furthermore due to the sign changing nonlinearities, the asymptotic sign condition is also not satisfied, causing further difficulties in proving the existence of global attractor. These difficulties are circumvented via the use of the lyapunov functional constructed along with the use of the uniform gronwall lemma. We are able to prove the existence of an attractor for the system, improving previous results in the literature.The Hausdorff and fractal dimensions of the attractor are also shown to be finite.In particular we derive a lower bound on the Hausdorff dimension of the global attractor. We use numerical simulations,as well as numerical attractor reconstruction methods via nonlinear time series analysis, to validate our results.
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