Existence and blowup of solutions for nonlinear evolution equations and their numerical computations

非线性演化方程解的存在性、爆炸性及其数值计算

• 批准号: RGPIN-2019-05940
• 负责人: Chen, Shaohua
• 金额: $ 1.24万
• 依托单位: Cape Breton University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=738607
• 关键词: Existence; blowup; solutions; nonlinear; evolution

Nonlinear evolution equations have many applications in descriptions of various models, such as reaction-diffusion, activator-inhibitor, fluid and quantum mechanics and population biology. Many equations involve degenerate or singular terms and some kinds of blow-up properties which cause many challenging problems in global, blow-up and numerical computations. The objectives of this research program are to investigate the properties of global and blow-up solutions for nonlinear evolution equations both theoretically and numerically, including convergence to steady states. The expected results will include: 1.Introduce a new functional method to discuss the global and blow-up solutions for the compressible Euler equations with variable damping coefficient. Also study the asymptotic behaviors, blow-up rate and blow-up time and steady states to the equations. 2.Investigate the existence and blow-up of solutions to higher order nonlinear Schrodinger equations. 3.Modify the existing algorithms for moving mesh methods and other adaptive grid methods to numerically solve compressible Euler equations, higher order nonlinear Schrodinger equations and some complicated equations, such as the equations whose solutions blow up at space infinity. Also develop a moving mesh scheme to simulate asymptotic behaviours in an unbounded domain. 4.Deal with a class of more general quasi-linear parabolic and hyperbolic systems to find sufficient conditions on initial data to deduce global existence and blow-up properties both theoretically and numerically. The new functional method is a very powerful method to obtain a priori estimate for elliptic and parabolic equations and will be introduced to hyperbolic equations. In the functional method, we consider an integral of nth power of several solutions. Taking derivatives with respect to t and integrating by parts we obtain a differential inequality. To my knowledge, if super- and sub-solution methods can be applied to a system, then the functional method can also be applied to the system. However, the functional method only requires weaker conditions. To obtain a numerical solution in an unbounded domain, we first map the unbounded domain into a bounded domain and change equations with some kind of singularity. Then use moving mesh method to reduce errors near the singularity. The numerical solutions also serve as a guide showing when the solutions blow up, exist globally or approach a steady state.

非线性发展方程在描述各种模型中有着广泛的应用，如反应扩散模型、活化剂-抑制剂模型、流体力学和量子力学模型以及种群生物学模型。许多方程都含有退化项或奇异项以及某些爆破性质，这给整体解、爆破解和数值计算带来了许多挑战性的问题。本研究计划的目标是从理论和数值两方面研究非线性发展方程的整体解和爆破解的性质，包括收敛到稳态。本文的主要成果包括：1.引入一种新的泛函方法，讨论变阻尼可压缩Euler方程的整体解和爆破解。研究了方程的渐近性态、爆破速率、爆破时间和定态。2.研究高阶非线性薛定谔方程解的存在性和爆破。3.改进现有的移动网格方法和其他自适应网格方法，使其能够数值求解可压缩Euler方程、高阶非线性Schrodinger方程以及一些复杂的方程，如解在空间无穷远爆破的方程。还开发了一个移动网格计划来模拟无界域中的渐近行为。4.处理一类更一般的拟线性抛物和双曲方程组，找到关于初始值的充分条件，从理论上和数值上推导出整体解的存在性和爆破性质。新的泛函方法是一个非常强大的方法来获得先验估计椭圆和抛物型方程，并将被引入到双曲型方程。在泛函方法中，我们考虑多个解的n次幂积分。对t求导并分部积分，得到一个微分不等式。据我所知，如果超解和次解方法可以应用于一个系统，那么函数方法也可以应用于这个系统。然而，泛函方法只需要较弱的条件。为了得到无界区域的数值解，我们首先将无界区域映射到有界区域，并将方程变换为具有某种奇异性。然后采用移动网格法减小奇异点附近的误差。数值解也可以作为一个指南，显示解何时爆破，全局存在或接近稳定状态。