Dirichlet problems associated to abstract nonlocal space-time differential operators
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 derive a 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.
- Publication:
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arXiv e-prints
- Pub Date:
- April 2024
- DOI:
- 10.48550/arXiv.2404.11289
- arXiv:
- arXiv:2404.11289
- Bibcode:
- 2024arXiv240411289W
- Keywords:
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- Mathematics - Analysis of PDEs;
- 35R11;
- 35E15 (Primary) 47D06;
- 47A60 (Secondary)
- E-Print:
- 19 pages