Dirichlet problems associated to abstract nonlocal space-time differential operators
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916675413082112 |
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| author | Willems, Joshua |
| author_facet | Willems, Joshua |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_11289 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dirichlet problems associated to abstract nonlocal space-time differential operators Willems, Joshua Analysis of PDEs 35R11, 35E15 (Primary) 47D06, 47A60 (Secondary) 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. |
| title | Dirichlet problems associated to abstract nonlocal space-time differential operators |
| topic | Analysis of PDEs 35R11, 35E15 (Primary) 47D06, 47A60 (Secondary) |
| url | https://arxiv.org/abs/2404.11289 |