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Variational Solutions to Nonlinear Diffusion Equations with Singular Diffusivity

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Abstract

We provide existence results for nonlinear diffusion equations with multivalued time-dependent nonlinearities related to convex continuous not coercive potentials. The results in this paper, following a variational principle, state that a generalized solution of the nonlinear equation can be retrieved as a solution of an appropriate minimization problem for a convex functional involving the potential and its conjugate. In the not coercive case, this assertion is conditioned by the validity of a relation between the solution and the nonlinearity. A sufficient condition, under which this relation is true, is given. At the end, we present a discussion on the solution existence for a particular equation describing a self-organized criticality model.

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Acknowledgements

This work was supported by a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0027.

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Authors and Affiliations

  1. Institute of Mathematical Statistics and Applied Mathematics, Romanian Academy, Calea 13 Septembrie 13, Bucharest, Romania

    Gabriela Marinoschi

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  1. Gabriela Marinoschi
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Correspondence to Gabriela Marinoschi.

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Communicated by Mimmo Iannelli.

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Marinoschi, G. Variational Solutions to Nonlinear Diffusion Equations with Singular Diffusivity. J Optim Theory Appl 161, 430–445 (2014). https://doi.org/10.1007/s10957-013-0430-5

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  • DOI: https://doi.org/10.1007/s10957-013-0430-5

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