June  2016, 9(3): 675-685. doi: 10.3934/dcdss.2016021

$1$-dimensional Harnack estimates

1. 

Hacettepe University, 06800, Beytepe, Ankara, Turkey

2. 

Dipartimento di Matematica "F. Casorati”, Università di Pavia, Via Ferrata 1, 27100 Pavia

3. 

Dipartimento di Matematica e Informatica "U. Dini", Università di Firenze, viale Morgagni, 67/A, 50134, Firenze

Received  March 2015 Revised  July 2015 Published  April 2016

Let $u$ be a non-negative super-solution to a $1$-dimensional singular parabolic equation of $p$-Laplacian type ($1< p <2$). If $u$ is bounded below on a time-segment $\{y\}\times(0,T]$ by a positive number $M$, then it has a power-like decay of order $\frac p{2-p}$ with respect to the space variable $x$ in $\mathbb R\times[T/2,T]$. This fact, stated quantitatively in Proposition 1.2, is a ``sidewise spreading of positivity'' of solutions to such singular equations, and can be considered as a form of Harnack inequality. The proof of such an effect is based on geometrical ideas.
Citation: Fatma Gamze Düzgün, Ugo Gianazza, Vincenzo Vespri. $1$-dimensional Harnack estimates. Discrete & Continuous Dynamical Systems - S, 2016, 9 (3) : 675-685. doi: 10.3934/dcdss.2016021
References:
[1]

E. DiBenedetto, Degenerate Parabolic Equations,, Universitext, (1993). doi: 10.1007/978-0-387-94020-5. Google Scholar

[2]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equation,, Acta Mathematica, 200 (2008), 181. doi: 10.1007/s11511-008-0026-3. Google Scholar

[3]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations,, Springer Monographs in Mathematics, (2012). doi: 10.1007/978-1-4614-1584-8. Google Scholar

[4]

E. DiBenedetto, U. Gianazza and V. Vespri, A New Approach to the Expansion of Positivity Set of Non-negative Solutions to Certain Singular Parabolic Partial Differential Equations,, Proc. Amer. Math. Soc., 138 (2010), 3521. doi: 10.1090/S0002-9939-2010-10525-7. Google Scholar

[5]

F. G. Düzgün, P. Marcellini and V. Vespri, An alternative approach to the Hoelder continuity of solutions to some elliptic equations,, Nonlinear Anal., 94 (2014), 133. doi: 10.1016/j.na.2013.08.018. Google Scholar

[6]

F. G. Düzgün, P. Marcellini and V. Vespri, Space expansion for a solution of an anisotropic $p$-Laplacian equation by using a parabolic approach,, Riv. Mat. Univ. Parma, 5 (2014), 93. Google Scholar

[7]

T. Kuusi, Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 7 (2008), 673. doi: 10.2422/2036-2145.2008.4.04. Google Scholar

[8]

V. Liskevich and I. I. Skrypnik, Hölder continuity of solutions to an anisotropic elliptic equation,, Nonlinear Anal., 71 (2009), 1699. doi: 10.1016/j.na.2009.01.007. Google Scholar

show all references

References:
[1]

E. DiBenedetto, Degenerate Parabolic Equations,, Universitext, (1993). doi: 10.1007/978-0-387-94020-5. Google Scholar

[2]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack estimates for quasi-linear degenerate parabolic differential equation,, Acta Mathematica, 200 (2008), 181. doi: 10.1007/s11511-008-0026-3. Google Scholar

[3]

E. DiBenedetto, U. Gianazza and V. Vespri, Harnack's Inequality for Degenerate and Singular Parabolic Equations,, Springer Monographs in Mathematics, (2012). doi: 10.1007/978-1-4614-1584-8. Google Scholar

[4]

E. DiBenedetto, U. Gianazza and V. Vespri, A New Approach to the Expansion of Positivity Set of Non-negative Solutions to Certain Singular Parabolic Partial Differential Equations,, Proc. Amer. Math. Soc., 138 (2010), 3521. doi: 10.1090/S0002-9939-2010-10525-7. Google Scholar

[5]

F. G. Düzgün, P. Marcellini and V. Vespri, An alternative approach to the Hoelder continuity of solutions to some elliptic equations,, Nonlinear Anal., 94 (2014), 133. doi: 10.1016/j.na.2013.08.018. Google Scholar

[6]

F. G. Düzgün, P. Marcellini and V. Vespri, Space expansion for a solution of an anisotropic $p$-Laplacian equation by using a parabolic approach,, Riv. Mat. Univ. Parma, 5 (2014), 93. Google Scholar

[7]

T. Kuusi, Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations,, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5), 7 (2008), 673. doi: 10.2422/2036-2145.2008.4.04. Google Scholar

[8]

V. Liskevich and I. I. Skrypnik, Hölder continuity of solutions to an anisotropic elliptic equation,, Nonlinear Anal., 71 (2009), 1699. doi: 10.1016/j.na.2009.01.007. Google Scholar

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