不定常区间上的发展方程
Evolution equations on time-dependent intervals
摘要：
研究了一类在不定常含时区间$l_1(t)<x<l_2(t)$，$0<t<T$上的线性发展型偏微分方程的初边值问题，其中$l_1(t)$和$l_2(t)$是实可微函数，$T$是任意常数.对于这样的问题，我们展示了如何在给定的初始和边界条件的未知边界值。作为说明性的例子，我们考虑热方程和线性薛定谔方程。在第一种情况下，未知的Neumann边界值通过具有显式核的两个线性积分方程的系统的唯一解表示为Dirichlet边界值和初始值。在第二种情况下，可以证明类似的结果，但仅适用于限制性更强的一类边界曲线。
英文摘要：
We study initial boundary value problems for linear evolution partial differential equations (PDEs) posed on a time-dependent interval $l_1(t)<x<l_2(t)$, $0<t<T$, where $l_1(t)$ and $l_2(t)$ are given, real, differentiable functions, and $T$ is an arbitrary constant. For such problems, we show how to characterise the unknown boundary values in terms of the given initial and boundary conditions. As illustrative examples we consider the heat equation and the linear Schrödinger equation. In the first case, the unknown Neumann boundary values are expressed in terms of the Dirichlet boundary values and of the initial value through the unique solution of a system of two linear integral equations with explicit kernels. In the second case, a similar result can be proved but only for a more restrictive class of boundary curves.}
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