Initial boundary value problems for time-fractional evolution equations in Banach spaces
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| Format: | Preprint |
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2025
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| author | Floridia, Giuseppe Golgeleyen, Fikret Yamamoto, Masahiro |
| author_facet | Floridia, Giuseppe Golgeleyen, Fikret Yamamoto, Masahiro |
| contents | We consider an initial value problem for time-fractional evolution equation in Banach space $X$: $$ \pppa (u(t)-a) = Au(t) + F(t), \quad 0<t<T. \eqno{(*)} $$ Here $u: (0,T) \rrrr X$ is an $X$-valued function defined in $(0,T)$, and $a \in X$ is an initial value. The operator $A$ satisfies a decay condition of resolvent which is common as a generator of analytic semigroup, and in particular, we can treat a case $X=L^p(\OOO)$ over a bounded domain $\OOO$ and a uniform elliptic operator $A$ within our framework.
First we construct a solution operator $(a, F) \rrrr u$ by means of $X$-valued Laplace transform, and we establish the well-posedness of (*) in classes such as weak solution and strong solutions. We discuss also mild solutions local in time for semilinear time-fractional evolution equations. Finally we apply the result on the well-posedness to an inverse problem of determining an initial value and we establish the uniqueness for the inverse problem. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_06554 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Initial boundary value problems for time-fractional evolution equations in Banach spaces Floridia, Giuseppe Golgeleyen, Fikret Yamamoto, Masahiro Analysis of PDEs 34G10, 34G20, 35R30 We consider an initial value problem for time-fractional evolution equation in Banach space $X$: $$ \pppa (u(t)-a) = Au(t) + F(t), \quad 0<t<T. \eqno{(*)} $$ Here $u: (0,T) \rrrr X$ is an $X$-valued function defined in $(0,T)$, and $a \in X$ is an initial value. The operator $A$ satisfies a decay condition of resolvent which is common as a generator of analytic semigroup, and in particular, we can treat a case $X=L^p(\OOO)$ over a bounded domain $\OOO$ and a uniform elliptic operator $A$ within our framework. First we construct a solution operator $(a, F) \rrrr u$ by means of $X$-valued Laplace transform, and we establish the well-posedness of (*) in classes such as weak solution and strong solutions. We discuss also mild solutions local in time for semilinear time-fractional evolution equations. Finally we apply the result on the well-posedness to an inverse problem of determining an initial value and we establish the uniqueness for the inverse problem. |
| title | Initial boundary value problems for time-fractional evolution equations in Banach spaces |
| topic | Analysis of PDEs 34G10, 34G20, 35R30 |
| url | https://arxiv.org/abs/2502.06554 |