论文标题
非本地性动力学的收敛问题
Convergence problems in nonlocal dynamics with nonlinearity
论文作者
论文摘要
我们发现梯度结构以研究非局部非线性动力学系统中溶液的收敛性。主要但不是只有唯一的，我们使用lojasiewicz不等式来证明收敛导致具有连续或离散的时间域以及有限或无限维度空间域的各种空间。更具体地说，我们分析了Lotka-Volterra型动力学和浓度分散动力学。 Lotka-Volterra方程描述了一组物种的种群动态。在假设物种之间的相互作用是对称的假设下，我们提出了两种不同的方法来得出收敛结果。一种是熵诱捕方法，是为了适应Akin和Hofbauer的想法（Math。Biosci。61（1982）51-62），利用能量的单调性来绑定熵，这提供了溶液与所需平衡的近端距离。另一种受Jabin和Liu在（非线性30（2017）4220）观察的启发的方法是改变变量以解决梯度结构的奇异性质。我们将此想法应用于广义的Lokta-Volterra系统，例如正则化Lotka-Volterra系统和具有突变的无限尺寸Lotka-Volterra方程。浓度 - 分散动力学是一种新型方程式，灵感来自孤立波形的固定点公式。作为PETVIASHVILI迭代的连续时间类似物，我们旨在动态计算非平凡的单独波轮廓。使用梯度结构，我们推断出非平凡的孤立和周期性波轮廓的存在。不幸的是，收敛结果保持开放。取而代之的是，我们通过添加非线性扩散项并通过使用Lojasiewicz收敛框架来证明解决方案的收敛性来使浓度分散方程正规化。
We uncover the gradient structure to investigate the convergence of solutions in nonlocal nonlinear dynamical systems. Mainly but not exclusively, we use the Lojasiewicz inequality to prove convergence results in various spaces with continuous, or discrete temporal domain, and finite, or infinite dimensional spatial domain. To be more specific, we analyze Lotka-Volterra type dynamics and concentration-dispersion dynamics. Lotka-Volterra equations describe the population dynamics of a group of species. Under the assumption that the interaction between species is symmetric, we present two different methods to derive the convergence result. One, the entropy trapping method, is to adapt the idea of Akin and Hofbauer (Math. Biosci. 61 (1982) 51-62) of using monotonicity of the energy to bound the entropy, which provides the proximal distance of the solution from the desired equilibrium. Another method, inspired by Jabin and Liu's observation in (Nonlinearity 30 (2017) 4220) is to change variables to resolve the singular nature of gradient structure. We apply this idea to show the convergence result in generalized Lokta-Volterra systems, such as regularized Lotka-Volterra systems and infinite dimensional Lotka-Volterra equations with mutation. Concentration-dispersion dynamics is a new type of equation that is inspired by fixed point formulations for solitary wave shapes. As a continuous time analogue of Petviashvili iteration, we aim to dynamically calculate the nontrivial solitary wave profile. Using the gradient structure we deduce the existence of nontrivial solitary and periodic wave profiles. Unfortunately, the convergence result remains open. Instead, we regularize the concentration dispersion equations by adding a nonlinear diffusive term, and prove the convergence of the solutions by using the Lojasiewicz convergence framework.
