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Variational Solutions to Nonlinear Diffusion Equations with Singular Diffusivity

Author

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  • Gabriela Marinoschi

    (Romanian Academy)

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.

Suggested Citation

  • Gabriela Marinoschi, 2014. "Variational Solutions to Nonlinear Diffusion Equations with Singular Diffusivity," Journal of Optimization Theory and Applications, Springer, vol. 161(2), pages 430-445, May.
    • Handle: RePEc:spr:joptap:v:161:y:2014:i:2:d:10.1007_s10957-013-0430-5
    DOI: 10.1007/s10957-013-0430-5
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    References listed on IDEAS

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    1. Bántay, Péter & Jánosi, Imre M., 1992. "Self-organization and anomalous diffusion," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 185(1), pages 11-18.
    2. Viorel Barbu, 2012. "Optimal Control Approach to Nonlinear Diffusion Equations Driven by Wiener Noise," Journal of Optimization Theory and Applications, Springer, vol. 153(1), pages 1-26, April.
    3. Gabriela Marinoschi, 2012. "Existence of Solutions to Time-Dependent Nonlinear Diffusion Equations via Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 154(3), pages 792-817, September.
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