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

非线性演化方程在各种模型的描述中有着广泛的应用，如反应-扩散、激活剂-抑制剂、流体力学和量子力学以及种群生物学。许多方程涉及退化项或奇异项以及一些爆破性质，这在全局计算、爆破计算和数值计算中引起了许多具有挑战性的问题。本研究计划的目的是研究非线性演化方程的全局解和爆破解的理论和数值性质，包括收敛到稳态。预期结果将包括：1。引入一种新的泛函方法来讨论变阻尼系数可压缩欧拉方程的整体解和爆破解。同时研究了该方程的渐近性质、爆破速率、爆破时间和稳态。2. 研究高阶非线性薛定谔方程解的存在性和爆破性。3. 改进现有的移动网格法和其他自适应网格法的算法，对可压缩欧拉方程、高阶非线性薛定谔方程和一些复杂方程（如解在空间无穷远处爆炸的方程）进行数值求解。还开发了一种移动网格方案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型和双曲型系统，从理论上和数值上寻找初始数据上的充分条件来推导整体存在性和爆破性质。这种新的泛函方法是求解椭圆型和抛物型方程先验估计的一种非常有效的方法，将被引入到双曲型方程中。在泛函方法中，我们考虑若干解的n次积分。对t求导，分部积分我们得到一个微分不等式。据我所知，如果上解法和子解法可以应用于系统，那么函数法也可以应用于系统。然而，函数方法只需要较弱的条件。为了得到无界域上的数值解，我们首先将无界域映射到有界域上，并用某种奇点变换方程。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，说明解何时爆炸，何时全局存在或何时接近稳态

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.**