Electronic Journal of Differential Equations, Vol. 2025 (2025), No. 18, pp. 1-40.

Title: Quantitative estimates of L^p maximal regularity for nonautonomous operators and global existence for quasilinear equations

Authors: Theo Belin (Univ. Paris-Saclay, France)
         Pauline Lafitte (Univ. Paris-Saclay, France)
  
Abstract:
In this work, we obtain quantitative estimates of the continuity constant 
for the $L^p$ maximal regularity of relatively continuous nonautonomous 
operators $A : I \to \mathcal{L}(D,X)$, where $D \hookrightarrow X$ 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 
$u \mapsto \mathbb{A}(u)$. The estimates obtained rely on the precise 
asymptotic analysis of the continuity constant with respect to 
perturbations of the operator of the form $A(\cdot) + \lambda \text{Id}$ as 
$\lambda \to \pm \infty$. 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. 


Submitted October 8, 2024. Published February 26, 2025
Math Subject Classifications: 35K90, 35K59, 35D35, 47D06.
Key Words: Nonautonomous Cauchy problem; $L^p$ maximal regularity; relatively continuous operators; quasilinear equations.