Study of various problems in mathematical physics

在线性偏微分方程的Gevrey类解的延拓问题上，我们发现了用特征和Gevrey指数的多重性来描述延拓可能性的新结果。在解的渐近性质上，我们得到了关于次指数全局解存在性的Liouville定理的推广。在某些细菌的模式形成问题上，我们提出了一个数学模型，并通过计算机模拟恢复了已知的二维模式，发现它们来自相变。在双曲型方程反问题的唯一性上，给出了在系数较弱条件下的最佳Lip-schitz稳定性。在平面图形的重建问题中，提出了一种高效的重建算法，并证明了该算法在特殊情况下的稳定性。在非唯一的情况下，我们引入了各种权重函数，并找到了有趣的模式作为重建图像最大化它们。对于Reissner-Mindlin板模型，我们采用了新的稳定混合矩形有限元，并表明在薄极限上它趋向Kirchhoff模型而不引起锁紧。对于湍流中的壳模型，我们考察了符合相同相似律的混沌解，得到了它的Lyapunov谱。我们发现它们只在某些典型波数中含有显著的分量，因此我们挑选出一个大吸引子维的渐近表达式，并验证了它与数值计算的良好一致性。构造了尺度变换下积分算子不变的双正交小波，并给出了其应用。

On continuation of Gevrey class solutions of linear partial differential equations, we found new results describing the possibility of continuation in terms of the multiplicity of the characteristic and Gevrey index.On the asymptotic behavior of solutions, we found a generalization of Liouville's theorem to the existence of infra-exponential global solutions. In problem of pattern formation of some bacteria, we presented a mathematical model and via computer simulation we recovered the known 2D patterns and found that they come from phase transition.On the uniquensee of the inverse problems for hyperbolic equations, we presented a best Lip-schitz stability under weaker conditions on the coefficients. In the reconstruction problem of plane figures, we presented efficient reconstruction algorithm and showed its stability in unique case. In non-unique case, we introduced various weight functions and found interesting patterns as reconstruction images maximizing them.For the Reissner-Mindlin plate model we exploited new stable mixed rectangular finite element and showed that on the thin limit it goes to the Kirchhoff model without causing locking. On the shell model in turbulence, we examined chaos solution obeying the same similarity law and obtained its Lyapunov spectra. Finding that they contain components of significant magnitude only at some typical wave numbers, we singled out an asymptotic expression for the large attrac-tor dimension, and verified its good concordance with numerical calculation. We constructed bi-orthogonal wavelets adapted to integral operators invariant under scale transformation, and gave its applications.