Dirichlet problems associated to abstract nonlocal space-time differential operators

Let the abstract fractional space-time operator $({\mathrm{\partial }}_t+A{)}^s$ be given, where $s\in (0,\mathrm{\infty })$ and $-A\text{\hspace{0.17em}⁣}:\mathsf{D}(A)\subseteq X\to X$ is a linear operator generating a uniformly bounded strongly measurable semigroup $(S(t){)}_{t\ge 0}$ on a complex Banach space $X$. We consider the corresponding Dirichlet problem of finding a function $u\text{\hspace{0.17em}⁣}:\mathbb{R}\to X$ such that $({\mathrm{\partial }}_t+A{)}^{s}u(t)=0$ on $(t_0,\mathrm{\infty })$ and $u(t)=g(t)$ on $(-\mathrm{\infty },t_0]$, for given $t_0\in \mathbb{R}$ and $g\text{\hspace{0.17em}⁣}:(-\mathrm{\infty },t_0]\to X$. We define the concept of $L^p$-solutions, to which we associate a mild solution formula which expresses $u$ in terms of $g$ and $(S(t){)}_{t\ge 0}$ and generalizes the well-known variation of constants formula for the mild solution to the abstract Cauchy problem $u^{\mathrm{\prime }}+Au=0$ on $(t_0,\mathrm{\infty })$ 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.