Dirichlet problems associated to abstract nonlocal spacetime differential operators
Abstract
Let the abstract fractional spacetime 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 wellknown 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 RiemannLiouville and Caputo type initial value problems.
 Publication:

arXiv eprints
 Pub Date:
 April 2024
 DOI:
 10.48550/arXiv.2404.11289
 arXiv:
 arXiv:2404.11289
 Bibcode:
 2024arXiv240411289W
 Keywords:

 Mathematics  Analysis of PDEs;
 35R11;
 35E15 (Primary) 47D06;
 47A60 (Secondary)
 EPrint:
 19 pages