Quantitative estimates of $L^p$ maximal regularity for nonautonomous operators and global existence for quasilinear equations
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909133346701312 |
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| author | Belin, Théo Lafitte, Pauline |
| author_facet | Belin, Théo Lafitte, Pauline |
| contents | In this work, we obtain quantitative estimates of the continuity constant for the $L^p$ maximal regularity of relatively continuous nonautonomous operators $\mathbb{A} : I \longrightarrow \mathcal{L}(D,X)$, where $D \subset X$ densely and compactly. They allow in particular to establish a new general growth condition for the global existence of strong solutions of Cauchy problems for nonlocal quasilinear equations for a certain class of nonlinearities $u \longrightarrow \mathbb{A}(u)$. The estimates obtained rely on the precise asymptotic analysis of the continuity constant with respect to perturbations of the operator of the form $\mathbb{A}(\cdot) + λI$ as $λ\longrightarrow \pm \infty$. A complementary work in preparation supplements this abstract inquiry with an application of these results to nonlocal parabolic equations in noncylindrical domains depending on the time variable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_00386 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantitative estimates of $L^p$ maximal regularity for nonautonomous operators and global existence for quasilinear equations Belin, Théo Lafitte, Pauline Functional Analysis In this work, we obtain quantitative estimates of the continuity constant for the $L^p$ maximal regularity of relatively continuous nonautonomous operators $\mathbb{A} : I \longrightarrow \mathcal{L}(D,X)$, where $D \subset X$ densely and compactly. They allow in particular to establish a new general growth condition for the global existence of strong solutions of Cauchy problems for nonlocal quasilinear equations for a certain class of nonlinearities $u \longrightarrow \mathbb{A}(u)$. The estimates obtained rely on the precise asymptotic analysis of the continuity constant with respect to perturbations of the operator of the form $\mathbb{A}(\cdot) + λI$ as $λ\longrightarrow \pm \infty$. A complementary work in preparation supplements this abstract inquiry with an application of these results to nonlocal parabolic equations in noncylindrical domains depending on the time variable. |
| title | Quantitative estimates of $L^p$ maximal regularity for nonautonomous operators and global existence for quasilinear equations |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2403.00386 |