Maximal regularity for semilinear non-autonomous evolution equations in temporally weighted spaces

Author:

Hossni TebbaniORCID,Mahdi Achache

Abstract

AbstractWe consider the problem of maximal regularity for the semilinear non-autonomous evolution equations $$\begin{aligned} u'(t)+A(t)u(t)=F(t,u),\, t \text {-a.e.}, \, u(0)=u_0. \end{aligned}$$ u ( t ) + A ( t ) u ( t ) = F ( t , u ) , t -a.e. , u ( 0 ) = u 0 . Here, the time-dependent operators A(t) are associated with (time dependent) sesquilinear forms on a Hilbert space $$\mathcal {H}.$$ H . We prove the maximal regularity result in temporally weighted $$L^2$$ L 2 -spaces and other regularity properties for the solution of the previous problem under minimal regularity assumptions on the forms, the initial value $$u_0$$ u 0 and the inhomogeneous term F. Our results are motivated by boundary value problems.

Publisher

Springer Science and Business Media LLC

Subject

General Mathematics

Reference16 articles.

1. Achache, M.: Maximal regularity for the damped wave equations. J. Elliptic Parabol. Equ. 6, 835–870 (2020)

2. Achache, M.; Ouhabaz, E.M.: Non-autonomous right and left multiplicative perturbations and maximal regularity. Stud. Math. 242(1), 1–30 (2018)

3. Achache, M.; Ouhabaz, E.M.: Lions’ maximal regularity problem with $$H^{1/2}$$-regularity in time. J. Differ. Equ. 266, 3654–3678 (2019)

4. Achache, M.; Tebbani, H.: Non-autonomous maximal regularity in weighted space. Electron. J. Differ. Equ. 124, 1–24 (2020)

5. Achache, M.: Non autonomous maximal regularity for the fractional evolution equations, 2020. To appear in journal of evolution equations. hal-0-2927759 (2020)

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