Dirichlet problems associated to abstract nonlocal space-time differential operators

TL;DR

This work develops a rigorous framework for Dirichlet problems associated to abstract nonlocal space–time operators of the form $(\partial_t + A)^s$ on a Banach space $X$. By leveraging strongly measurable semigroups, fractional parabolic calculus, and the Phillips functional calculus, it defines $L^p$-solutions and introduces a novel mild solution formula that generalizes the classical variation-of-constants to non-integer orders $s$, and recovers the RL/Caputo formulations in appropriate limits. The paper proves that $L^p$-solutions imply mild solutions under natural assumptions, and provides explicit integer- and fractional-order representations, including an operator-valued incomplete gamma framework for special data. It also offers a detailed comparison with RL and Caputo Cauchy problems, highlighting the continuity advantages of the Dirichlet-mild formulation and clarifying how initial-data prescriptions differ across these nonlocal models. The results establish a robust, general theory for nonlocal space–time Dirichlet problems with potential applications to fractional SPDEs and related probabilistic interpretations.

Abstract

Let the abstract fractional space-time operator $(\partial_t + A)^s$ be given, where $s \in (0,\infty)$ and $-A \colon \mathsf{D}(A) \subseteq X \to X$ is a linear operator generating a uniformly bounded strongly measurable semigroup $(S(t))_{t\ge0}$ on a complex Banach space $X$. We consider the corresponding Dirichlet problem of finding a function $u \colon \mathbb{R} \to X$ such that $(\partial_t + A)^s u(t) = 0$ on $(t_0, \infty)$ and $u(t) = g(t)$ on $(-\infty, t_0]$, for given $t_0 \in \mathbb{R}$ and $g \colon (-\infty,t_0] \to X$. We define the concept of $L^p$-solutions, to which we associate a mild solution formula which expresses $u$ in terms of $g$ and $(S(t))_{t\ge0}$ and generalizes the well-known variation of constants formula for the mild solution to the abstract Cauchy problem $u' + Au = 0$ on $(t_0, \infty)$ with $u(t_0) = x \in \overline{\mathsf{D}(A)}$. Moreover, we include a comparison to analogous solution concepts arising from Riemann-Liouville and Caputo type initial value problems.

Theorems & Definitions (40)

• Definition 2.2: Strong solution
• Definition 2.3: Mild solution
• Proposition 2.4
• proof
• Definition 2.5: $L^p$-solution
• Proposition 2.6
• proof
• Remark 2.7
• Remark 2.8
• Definition 3.1: $L^p$-solution