On non-autonomous maximal L p -regularity under Besov regularity in time in weighted spaces

Mahdi Achache (1)

1 I2M - Institut de Mathématiques de Marseille

We consider the problem of maximal regularity for non-autonomous Cauchy problems u ′ (t) + A(t)u(t) = f (t), t-a.e., u(0) = u 0. The time dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space H. We prove the maximal regularity result in temporally weighted L p-spaces for p > 2 and other regularity properties for the solution of the previous problem under minimal regularity assumptions on the forms and the initial value u 0. Our main assumption is that (A(t)) t∈[0,τ ] are in the Besov space B 1− 1 p ,2 p with respect to the variable t and u 0 ∈ (H; D(A(0))) θ,p , where θ = p−1−β p. Our results are motivated by boundary value problems.

Mots clés: Besov spaces; maximal regularity; non-autonomous evolution equations; sesquilinear forms. weighted spaces

Domaines: Mathématiques [math]

Mahdi Achache. On non-autonomous maximal L p -regularity under Besov regularity in time in weighted spaces. 2021. ⟨hal-03350118⟩