The global behavior of solutions of evolution equations in noncylindrical domain with time-moving boundaries

1)设空间维度为1 ~ 5。考虑球中的非线性球对称自主波动方程。我们证明了方程有无穷多个时间周期解。在证明中，我们在拉普拉斯特征值和周期上使用了丢芬图不等式。为此，我们利用贝塞尔函数零点的渐近展开式，详细研究了贝塞尔函数零点的数值性质。2)考虑了重悬杆线性方程组的IBVP问题。假设时间准周期力作用于弦。假设了串算子的特征值和拟周期的一般丢芬图条件，从而完全明确了IBVP解的概周期结构。为了证明这一说法，我们定义了良好匹配的函数空间，解决了这些函数空间中的特征函数问题，并构造了谱理论。3)考虑了具有周期运动边界的球中非线性球对称波动方程的BVP问题。4)证明了流形上测大地上的下列定理：定理：设M是一个实数解析黎曼流形同胚于2球。如果M的高斯曲率为正，则每个点的共轭轨迹由一个点组成或至少有4个顶点。

1) Let the space dimension be 1 to 5. Consider nonlinear sphere-symmetric autonomous wave equations in ball. We showed that the equations have infinitely many time-periodic solutions. In the proof we used the Diophantine inequalities on the eigenvalues of Laplacian and the periods. To this end we studied in detail numerical properties of the zero points of the Bessel functions, using the asymptotic expansions of the zero points.2) We considered IBVP for linear equations of heavy suspended string. Assume that time-quasiperiodic force works to the string. We assume the general Diophantine conditions on the eigenvalues of the string operator and the quasi perios Then we completely made clear the strunture of almost periodic structure of the solutions of IBVP. To show this statement, we defined well-matched function spaces, solved the eigenfunction problem in these function spaces and constructed the spectral theory.3) We considered BVP for nonlinear sphere-symmetric wave equations in ball with periodically moving boundaries. Then we showed the existence of periodic solutions of the BVP.4) The following theorem on the geodesics on manifolds is proved by M.Tanaka.Theorem : Let M be a real analytic Riemaniann manifold homeomorphic to a 2-sphere. If the Gaussian curvature of M is positive, then the conjugate locus of each point consists of a single point or has at least 4 cusps.