Theory and Applications of Evolution Equations with Nonlocal Conditions

我们研究了两种不同类型的演化方程，重点关注了常见的关键词“非局部条件”。【肌肉收缩模型】肌肉收缩是粗丝（肌凝蛋白）和细丝（肌动蛋白）之间相对滑动的结果。当所谓的交叉桥将肌凝蛋白连接到肌动蛋白上，并起到弹簧的作用时，这种滑动就发生了。肌肉收缩模型描述了连接的交叉桥的密度随时间的变化。由于收缩速度取决于连接的交叉桥的数量，因此模型中出现了一个非局部项。在本研究中，我们考虑了双曲型输运方程和抛物型输运扩散方程。在允许交叉桥的连接和分离为非线性的一般情况下，得到了存在唯一性结果。[种群动态]基于个体大小和时间的种群模型作为描述森林或人工林中植物种群的自然模型进行了研究。《生育法》中有一个外地术语。得到了一类一般模型正解的存在性、爆破性和解的整体存在性的一些结果。1998年在美国纽波特海滩举行的国际会议上，我就这些结果做了一次演讲。此外，我们还展示了对出生和衰老函数以及初始数据的连续依赖。此外，我们还研究了具有非线性增长率的模型，并得到了存在性结果。1999年在德国柏林举行的一次国际会议上，我做了一个关于这个问题的演讲。

We have investigated two different types of evolution equations, focusing on the common key word "nonlocal conditions".[Muscle contraction models]The muscle contraction is a consequence of relative sliding between the thick filament (Myosin) and the thin filament (Actin).This sliding occurs when the so-called cross-bridges attach Myosin to Actins, and act as spring.The Muscle contraction model describes the temporal variation of the density of the attached cross-bridges.Since the contraction speed is dependent on the number of the attached cross-bridges, a nonlocal term appears in the model.In this research, we have considered transport equations of hyperbolic type and transport-diffusion equations of parabolic type.We obtained the existence and uniqueness results with some general setting allowing the attaching and detaching of the cross-bridges being nonlinear.[Population dynamics]The population model depending on the individuals size and time has been investigated as a natural model describing the population of plants in forests or plantations.The birth law includes a nonlocal term.We obtained some results concerning the existence of positive solutions and blow up and global existence of the solution to a general model.I had a talk on these results in the international conference held at Newport Beach, USA in 1998.Also, we have shown the continuous dependence on the birth and aging functions as well as initial data.Besides, we have investigated the model having the nonlinear growth rate and obtained the existence result.I had a talk on this in the international conference held at Berlin, Germany in 1999.