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Harnack estimates for non-negative weak solutions of a class of singular parabolic equations

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Abstract

We prove forward, backward and elliptic Harnack type inequalities for non-negative local weak solutions of singular parabolic differential equations of type

$$u_t={\rm div}{\bf A}(x, t, u, Du)$$

where A satisfies suitable structure conditions and a monotonicity assumption. The prototype is the parabolic p−Laplacian with 1 < p < 2. By using only the structure of the equation and the comparison principle, we generalize to a larger class of equations the estimates first proved by Bonforte et al. (Adv. Math. 224, 2151–2215, 2010) for the model equation.

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Authors and Affiliations

  1. Dipartimento di Matematica “F. Casorati”, Università degli Studi di Pavia, via Ferrata, 1, 27100, Pavia, Italy

    S. Fornaro

  2. Dipartimento di Matematica “U. Dini”, Università degli Studi di Firenze, viale Morgagni, 67/A, 50134, Firenze, Italy

    V. Vespri

Authors
  1. S. Fornaro
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  2. V. Vespri
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Correspondence to V. Vespri.

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Fornaro, S., Vespri, V. Harnack estimates for non-negative weak solutions of a class of singular parabolic equations. manuscripta math. 141, 85–103 (2013). https://doi.org/10.1007/s00229-012-0562-1

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  • DOI: https://doi.org/10.1007/s00229-012-0562-1

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