Analysis of Nonlinear Waves and Nonlinear Diffusions

在各种关于微分方程的问题中，在这个项目中，我们主要关注与应用数学、物理和技术相关的问题。总结1997-99年间研究人员取得的成果，可以说，该项目取得了丰硕的成果，发表了9篇论文，分析了非线性波和非线性扩散。(1)非线性波的衰减和渐近性：证明了具有非线性耗散的声波方程的散射态的存在性。建立了耗散在无穷远附近的描述弹性弦振动的Kirchhoff方程整体小解的存在性及其能量衰减性。(2)半线性或拟线性退化抛物型方程：考虑具有非线性功率源项的反应扩散系统。划分S…爆破和整体存在性的临界指数更多的解决方案被证明是存在的。此外，还证明了爆破解的临界爆破性、寿命以及时间的渐近性。对于描述多孔介质中的流体和等离子体中的燃烧过程的拟线性方程，也得到了类似的结果。(3)谱和散射理论：得到了经典Sturm-Liouville算子的谱反问题的新的公式和结果。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，我们扩展了极限吸收原理对于振荡长程势的薛定谔算子的适用性。每个研究者都发展了许多重要的非线性问题，其中包括：哈密顿-雅可比方程的齐次化，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。较少

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