广义指数有界积分半群
Generalized exponentially bounded integrated semigroups
AI总结：
本文针对带分布初值与右端项的柯西问题相关的指数有界积分半群序列，引入伴随等价关系刻画无穷小生成元序列与对应积分半群序列的依赖关系，完成两类序列的比较与对应确定。
摘要：
本文的核心研究对象是指数有界积分半群序列的分析，这类半群与柯西问题（方程见式\ref{jed}）相关：$\frac{\text{d}}{\text{d}t}u(t,x)-a(D)u(t,x)=f(t,x)$，初值$u(0,x)=u_0(x)$，$t\bigtriangleup0$，$x\bigtriangleup\bigtriangleup^d$，其中初值$u_0$和右端项$f$均为分布，通过一系列带正则化$u_0$与$f$、且用(伪)微分算子序列$a_n(D)$替代$a(D)$的方程建立关联。论文的核心内容是比较无穷小生成元序列，并确定对应的积分半群序列。为此，我们引入了伴随关系：一侧是无穷小生成元的等价关系，另一侧是积分半群的对应等价关系。对生成元所设假设的阶数本质上刻画了无穷小生成元序列与对应积分半群序列之间的相互依赖关系。
英文摘要：
The main subject of this paper is the analysis of sequences of exponentially bounded integrated semigroups which are related to Cauchy problems \begin{equation}\label{jed} \frac{\partial}{\partial t}u(t,x)-a(D)u(t,x)=f(t,x), \quad u(0,x)=u_0(x), \quad t\geq 0, \ x\in \mathbb R^d, \end{equation} with a distributional initial data $u_0$ and a distributional right hand side $f$ through a sequence of equations with regularized $u_0$ and $f$ and a sequence of (pseudo) differential operators $a_n(D)$ instead of $a(D)$. Comparison of sequences of infinitesimal generators and the determination of corresponding sequences of integrated semigroups are the main subject of the paper. For this purpose, we introduce association, the relation of equivalence for infinitesimal generators on one side and the corresponding relations of equivalence of integrated semigroups on another side. The order of involved assumptions on generators essentially characterize the mutual dependence of sequences of infinitesimal generators and the corresponding sequences of integrated semigroups.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
