On maximal regularity for the second order Cauchy problems in Hilbert space

En régulatrité maximale des equations d'order 2 dans l'espace de Hilbert.

Mahdi Achache (1)

1 IMB - Institut de Mathématiques de Bordeaux

We consider the problem of maximal regularity for semilinear non-autonomous second order Cauchy problems    u ′′ (t) + B(t)u ′ (t) + A(t)u(t) = F (t, u, u ′) t-a.e. u(0) = u 0 , u ′ (0) = u 1. (0.1) Here, the time dependent operator A(t) is bounded from the Hilbert space V to its dual space V ′ and B(t) is associated with a sesquilinear form b(t, ·, ·) with domain V. We prove maximal L 2-regularity results and other regularity properties for the solutions of the above equation under minimal regularity assumptions on the operators and the inhomogeneous term F. One of our main results shows that maximal L 2-regularity holds if the operators are piecewise H 1 2 with respect to t. This regularity assumption is optimal and provides the best positive result on this problem.

Mots clés: Damped wave equation; maximal regularity; non-autonomous evolution equations Mathematics Subject Classification (2010): 35K90; 35K45; 47D06

Domaines: Mathématiques [math]

Mahdi Achache. On maximal regularity for the second order Cauchy problems in Hilbert space. 2020. ⟨hal-02943852⟩