Quantitative estimates of $L^p$ maximal regularity for nonautonomous operators and global existence for quasilinear equations
- Resource Type
- Working Paper
- Authors
- Belin, Théo; Lafitte, Pauline
- Source
- Subject
- Mathematics - Functional Analysis
- Language
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 $\mathbb{A}(\cdot) + \lambda I$ as $\lambda \longrightarrow \pm \infty$. 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.