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

非线性发展方程在描述各种模型方面有着广泛的应用，如反应扩散、活化剂-抑制剂、流体和量子力学以及种群生物学等。许多方程涉及退化或奇异项和某些爆破性质，这在整体、爆破和数值计算中引起了许多具有挑战性的问题。本研究的目的是从理论和数值两个方面研究非线性发展方程整体解和爆破解的性质，包括收敛到定态。本文的预期结果包括：1.引入一种新的泛函方法来讨论变阻尼可压缩欧拉方程的整体解和爆破解。还研究了方程的渐近行为、爆破速度、爆破时间和稳态。2.研究了高阶非线性薛定谔方程解的存在性和爆破性质。3.对已有的移动网格法和其他自适应网格法进行了改进，使之能够数值求解可压缩欧拉方程、高阶非线性薛定谔方程和一些复杂的方程，如解在空间无穷远处爆破的方程。还开发了一种移动网格格式来模拟无界区域中的渐近行为。4.研究了一类更一般的拟线性抛物型和双曲型方程组，得到了关于初值的充分条件，从理论上和数值上证明了整体解的存在性和爆破性质。新的泛函方法是获得椭圆型和抛物型方程先验估计的一种非常有效的方法，并将被引入到双曲型方程中。在泛函方法中，我们考虑几个解的n次方积分。取关于t的导数，分部分积分，得到一个微分不等式。据我所知，如果上解和子解方法可以应用于一个系统，那么函数方法也可以应用于该系统。然而，泛函方法只需要较弱的条件。为了得到无界区域上的数值解，我们首先将无界区域映射到有界区域，并用某种奇性变换方程。然后用移动网格法减小奇点附近的误差。数值解还可以作为指导，显示解何时爆炸、全局存在或接近稳定状态。

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.