An Evolutionary Boundary Value Problem |
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Authors: | Benseghir Aissa Sofonea Mircea |
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Institution: | 1.Université Ferhat Abbas Sétif 1, El Bez, 19000, Sétif, Algeria ;2.Laboratoire de Mathématiques et Physique, University of Perpignan Via Domitia, 52 Avenue Paul Alduy, 66860, Perpignan Cedex, France ; |
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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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