高维非线性偏微分方程解的延迟奇点形成

Delayed singularity formation for solutions of nonlinear partial differential equations in higher dimensions.

作者信息

John F

机构信息

Courant Institute of Mathematical Sciences, New York University, New York, N.Y. 10012.

出版信息

Proc Natl Acad Sci U S A. 1976 Feb;73(2):281-2. doi: 10.1073/pnas.73.2.281.

DOI:10.1073/pnas.73.2.281

PMID:16578737

原文链接:https://pmc.ncbi.nlm.nih.gov/articles/PMC335889/

Abstract

Strict solutions u of genuinely nonlinear homogeneous hyperbolic equations in two independent variables with initial data f(x) of compact support become singular after a time interval of order parallelf parallel(-1). In higher dimensions solutions initially of compact support are likely to have life expectancies of orders parallelf parallel(-2+epsilon) at least. This is proved for the special case of solutions u(x(1),..., x(n), t) of a second order equation u(tt) = Sigma(i,j)a(ij)u(xixj), where n >/= 3 and where the coefficients a(ij) are C(infinity)-functions in the first derivatives of u, forming a symmetric positive definite matrix.

摘要

具有紧支集初始数据(f(x))的两个自变量的真正非线性齐次双曲方程的严格解(u)在阶数为(|f|^{-1})的时间间隔后会变得奇异。在更高维度中，最初具有紧支集的解的寿命预期至少为阶数(|f|^{-2 + \epsilon})。这在二阶方程(u_{tt} = \sum_{i,j}a_{ij}u_{x_ix_j})的解(u(x_1,\ldots,x_n,t))的特殊情况下得到证明，其中(n \geq 3)且系数(a_{ij})是(u)的一阶导数中的(C^{\infty})函数，形成一个对称正定矩阵。