Quantitative estimates of Lᵖ maximal regularity for nonautonomous operators and global existence for quasilinear equations

In this work, we obtain quantitative estimates of the continuity constant for the Lᵖ maximal regularity of relatively continuous nonautonomous operators A : I L(D,X), where D ⊂ 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 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(·) + λ I as λ ± ∞. 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.