Extended groups of semigroups and backward problems of heat equations

TL;DR

This paper investigates backward solvability of heat equations within an abstract framework, demonstrating that heat semigroups can be extended to groups in larger spaces under certain conditions.

Contribution

It introduces a method to extend heat semigroups to groups in larger Banach spaces, based on density and backward uniqueness assumptions.

Findings

Semigroups generated by heat operators are extendable to groups in an extended space.

Holomorphic semigroups satisfy the conditions for extension to groups.

Structural properties of the extended space are analyzed.

Abstract

In this paper, we are concerned with backward solvabilities of heat equations, in an abstract framework. We show that semigroups $T_t$ in Banach spaces $X$, generated by heat operators, are extendable to groups in an extended space $E$, which is obtained by considering a sequence of wider Banach spaces containing $X$, i.e. $X$$/subset$$X_t$$/subset$$X_s$... $(t<s)$, under the following two conditions. One is the density assumption on a subset $D$ of $X$, the set of initial values $x$ from which $T_{-t}x$ exists for all $t>0$. Another is the backward uniqueness of the semigroup $T_t$. For example, we prove the holomorphic semigroup satisfies the above conditions, and thus is extendable to a group in a larger functional space $E$. We also studied structual properties of the extended space $E$.