Study of various problems in mathematical physics

点击翻译按钮获取中文摘要

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.