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Degenerate Nonlinear Diffusion Equations pp 91–108Cite as

Existence for Nonautonomous Parabolic–Elliptic Degenerate Diffusion Equations

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Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2049))

Abstract

The subject of this chapter is the study of a Cauchy problem with a time-dependent nonlinear operator which is the abstract formulation of a boundary value problem for a fast diffusion equation in the parabolic–elliptic degenerate case, with nonhomogeneous Neumann conditions. Existence and uniqueness for the abstract Cauchy problem are proved in relation with the results of Kato, given in [71] and extended by Crandall and Pazy in [44] for the nonautonomous evolution equations with nonlinear m-accretive operators.

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References

  1. H. Brezis, I. Ekeland, Un principe variationnel associé à certaines equations paraboliques. Le cas independant du temps. C. R. Acad. Sci. Paris Sér. A 282, 971–974 (1976)

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  2. H. Brezis, I. Ekeland, Un principe variationnel associé à certaines equations paraboliques. Le cas dependant du temps. C. R. Acad. Sci. Paris Sér. A 282, 1197–1198 (1976)

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  3. M.G. Crandall, A. Pazy, Nonlinear evolution equations in Banach spaces. Isr. J. Math. 11, 57–94 (1971)

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  4. M.G. Crandall, A. Pazy, An approximation of integrable functions by step functions with an application. Proc. Amer. Math. Soc. 76(1), 74–80 (1979)

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  6. T. Kato, Nonlinear semi-groups and evolution equations. J. Math. Soc. Jpn. 19, 508–520 (1967)

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  7. G. Marinoschi, Existence to time-dependent nonlinear diffusion equations via convex optimization. JOTA 154, 3 (2012) (online, DOI: 10.1007/s10957-012-0017-6)

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Authors and Affiliations

  1. Department of Mathematics, University of Bologna, Bologna, Italy

    Angelo Favini

  2. Romanian Academy Institute of Mathematical Statistics and Applied Mathematics, Bucharest, Romania

    Gabriela Marinoschi

Authors
  1. Angelo Favini
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  2. Gabriela Marinoschi
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Favini, A., Marinoschi, G. (2012). Existence for Nonautonomous Parabolic–Elliptic Degenerate Diffusion Equations. In: Degenerate Nonlinear Diffusion Equations. Lecture Notes in Mathematics, vol 2049. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-28285-0_3

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