具有分布右端和间断系数的线性Cauchy问题的适定性
On well-posedness of the linear Cauchy problem with the distributional right-hand side and discontinuous coefficients
摘要：
本文证明了$x^{\prime}-A(t)x=f$型线性微分系统Cauchy问题的适定性，其中$f$是一个分布，$A$至多具有第一类间断，并且它的所有导数几乎处处定义.这个系统的左手边包含一个分布和一个不连续函数的乘积，一般来说，这个不连续函数在具有光滑检验函数$\mathcal D'$的经典分布空间中是不确定的，所以柯西问题在$\mathcal D'$中没有解。在下文中，我们考虑这个系统在空间的分布与不连续的检验函数，其元素允许连续和结合乘法的功能拥有最多的第一类不连续性（连同其所有衍生物定义几乎处处不在），并表明存在唯一的解决方案的柯西问题，连续依赖于$f$。
英文摘要：
We prove the well-posedness of the Cauchy problem for the linear differential system of the form $x^{\prime}-A(t)x=f$, where $f$ is a distribution and $A$ possesses at most first-kind discontinuities together with all its derivatives defined almost everywhere. The left-hand side of this system contains the product of a distribution and, in general, a discontinuous function, which is undefined in the classical space of the distributions with the smooth test functions $\mathcal D'$, so the Cauchy problem has no solution in $\mathcal D'$. In what follows, we cosider this system in the space of distributions with the discontinuous test functions, whose elements admit continuous and associative multiplication by functions possessing at most first-kind discontinuities (together with all their derivatives defined almost everywhere), and show that there exists the unique solution of the Cauchy problem which depends continuously on $f$.
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