Maximal Regularity for Non-Autonomous Second Order Cauchy Problems |
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Authors: | Dominik Dier El Maati Ouhabaz |
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Institution: | 1. Institute of Applied Analysis, University of Ulm, 89069, Ulm, Germany 2. Institut de Mathématiques (IMB), Univ. Bordeaux, 351, cours de la Libération, 33405, Talence cedex, France
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Abstract: | We consider non-autonomous wave equations $$\left\{\begin{array}{ll}\ddot{u}(t) + \mathcal{B}(t) \dot{u}(t) + \mathcal{A}(t)u(t) = f(t) \quad t{\text -}{\rm a.e.}\\ u(0) = u_{0},\, \dot{u}(0) = u_{1}.\\\end{array}\right.$$ where the operators ${\mathcal{A}(t)}$ and ${\mathcal{B}(t)}$ are associated with time-dependent sesquilinear forms ${\mathfrak{a}(t, ., .)}$ and ${\mathfrak{b}}$ defined on a Hilbert space H with the same domain V. The initial values satisfy ${u_0 \in V}$ and ${u_1 \in H}$ . We prove well-posedness and maximal regularity for the solution both in the spaces V′ and H. We apply the results to non-autonomous Robin-boundary conditions and also use maximal regularity to solve a quasilinear problem. |
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