Analysis of Nonlinear Waves and Nonlinear Diffusions

在各类微分方程问题中，本项目主要关注与应用数学、物理和技术相关的问题。总结1997 ~ 1999年的研究成果，可以说本课题的研究目标是圆满的，主要研究者发表了9篇关于非线性波和非线性扩散的论文。主要内容包括：（1）非线性波的衰减与渐近性：证明了具有非线性耗散的声波方程散射态的存在性。研究了一类具耗散项的弹性弦振动的Kirchhoff方程的整体小解的存在性及其能量衰减。(2)考虑半线性或拟线性退化抛物方程：具有非线性幂源项的反应扩散方程组。区分s的爆破和整体存在的临界指数 关于我们 解决方案是存在的。证明了爆破解的临界爆破性、爆破解的生存期以及整体解在时间趋于无穷远时的渐近性。对于描述多孔介质中流体和等离子体中燃烧过程的拟线性方程组，也得到了类似的结果。(3)光谱和散射理论：一个新的配方和结果，获得了经典的Sturm-Liouville算子的光谱逆问题。建立了具有小耗散或囤积的波动方程的散射理论。此外，通过推广前人的结果，我们扩展了薛定谔算符振荡长程势的限制吸收原理的适用范围。每个研究者发展了许多重要的非线性问题，其中包括，Hamilton-Jacobi方程的均匀化、Hele-Shaw流的自由边界问题、变分问题、Navier-Stokes方程的稳定性和非线性散射。少

Among various kind of problems on differential equations, in this project, we are mainly concerned with those related to the Applied Mathematics, Physics and Technology. Summarizing the results obtained by the investigators in the period 1997-99, we can say that the objective of this project is accomplished fruitfully.The head investigator published 9 papers analyzing the nonlinear waves and nonlinear diffusions. The topics include the following:(1) Decay and asymptotics of nonlinear waves: The existence of the scattering state is proved for acoustic wave equations with nonlinear dissipation. The existence of global small solution and its energy decay are established for the Kirchhoff equation (describing the vibration of elastic string) with dissipation localized near infinity.(2) Semilinear or quasilinear degenerate parabolic equations: Reaction diffusion systems with nonlinear power source term are considered. The critical exponents which divide the blow-up and global existence of s … More olutions are shown to exist. Moreover, the critical blow-up, life span of blow-up solutions and the asymptotic behavior for time goes to infinity of global solutions are proved. Similar results are also obtained for the quasilinear equations describing fluids in porous media and combustion process in plasma.(3) Spectral and scattering theory: A new formulation and results are obtained for the spectral inverse problem for the classical Sturm-Liouville operator. Scattering theory is established for the wave equation with small dissipation or hoarding. Moreover, by generalizing the former results, we expanded the applicability of the principle of limiting absorption for the Schrodinger operator oscillating long-range potential.Each investigator developed many important nonlinear problems, among which are included e.g., homogenization of the Hamilton-Jacobi equation, free boundary problem of Hele-Shaw flow, variational problems, stability of Navier-Stokes equations and nonlinear scattering. Less