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Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations

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Abstract

We prove the existence of suitably defined weak Radon measure-valued solutions of the homogeneous Dirichlet initial-boundary value problem for a class of strongly degenerate quasilinear parabolic equations. We also prove that: \((i)\) the concentrated part of the solution with respect to the Newtonian capacity is constant; \((ii)\) the total variation of the singular part of the solution (with respect to the Lebesgue measure) is nonincreasing in time. Conditions under which Radon measure-valued solutions of problem \((P)\) are in fact function-valued (depending both on the initial data and on the strength of degeneracy) are also given.

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Authors and Affiliations

  1. Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, P.le A. Moro 5, 00185 , Rome, Italy

    Maria Michaela Porzio, Flavia Smarrazzo & Alberto Tesei

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  1. Maria Michaela Porzio
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  2. Flavia Smarrazzo
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  3. Alberto Tesei
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Correspondence to Alberto Tesei.

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Communicated by L. Ambrosio.

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Porzio, M.M., Smarrazzo, F. & Tesei, A. Radon measure-valued solutions of nonlinear strongly degenerate parabolic equations. Calc. Var. 51, 401–437 (2014). https://doi.org/10.1007/s00526-013-0680-y

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