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Evolution equations governed by Lipschitz continuous non-autonomous forms

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Abstract

We prove L 2-maximal regularity of the linear non-autonomous evolutionary Cauchy problem

$$\dot u(t) + A(t)u(t) = f(t){\text{ for a}}{\text{.e}}{\text{. }}t \in \left[ {0,T} \right],{\text{ }}u(0) = {u_0}$$

, where the operator A(t) arises from a time depending sesquilinear form a(t, ·, ·) on a Hilbert space H with constant domain V. We prove the maximal regularity in H when these forms are time Lipschitz continuous. We proceed by approximating the problem using the frozen coefficient method developed by El-Mennaoui, Keyantuo, Laasri (2011), El-Mennaoui, Laasri (2013), and Laasri (2012). As a consequence, we obtain an invariance criterion for convex and closed sets of H.

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Authors and Affiliations

  1. Department of Mathematics, University Ibn Zohr, Faculty of Sciences, BP 32/S, 80000, Agadir, Morocco

    Ahmed Sani

  2. Fachbereich C-Mathematik und Naturwissenschaften, University of Wuppertal, Gaußstraße 20, 42097, Wuppertal, Germany

    Hafida Laasri

Authors
  1. Ahmed Sani
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  2. Hafida Laasri
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Correspondence to Ahmed Sani.

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Sani, A., Laasri, H. Evolution equations governed by Lipschitz continuous non-autonomous forms. Czech Math J 65, 475–491 (2015). https://doi.org/10.1007/s10587-015-0188-z

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  • DOI: https://doi.org/10.1007/s10587-015-0188-z

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