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An Evolutionary Boundary Value Problem

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Abstract

We consider a nonlinear initial boundary value problem in a two-dimensional rectangle. We derive variational formulation of the problem which is in the form of an evolutionary variational inequality in a product Hilbert space. Then, we establish the existence of a unique weak solution to the problem and prove the continuous dependence of the solution with respect to some parameters. Finally, we consider a second variational formulation of the problem, the so-called dual variational formulation, which is in a form of a history-dependent inequality associated with a time-dependent convex set. We study the link between the two variational formulations and establish existence, uniqueness, and equivalence results.

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Authors and Affiliations

  1. Université Ferhat Abbas Sétif 1, El Bez, 19000, Sétif, Algeria

    Aissa Benseghir

  2. Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860, Perpignan Cedex, France

    Mircea Sofonea

Authors
  1. Aissa Benseghir
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  2. Mircea Sofonea
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Correspondence to Mircea Sofonea.

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Benseghir, A., Sofonea, M. An Evolutionary Boundary Value Problem. Mediterr. J. Math. 13, 4463–4480 (2016). https://doi.org/10.1007/s00009-016-0756-y

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  • DOI: https://doi.org/10.1007/s00009-016-0756-y

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