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

$1$-dimensional Harnack estimates

Fatma Gamze Düzgün 1, , Ugo Gianazza 2, and  Vincenzo Vespri 3,

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.
Keywords: Singular diffusion equations, expansion of positivity., $p$-laplacian.
Mathematics Subject Classification: Primary: 35K65, 35B65; Secondary: 35B4.
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, Springer-Verlag, New York, 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), 181-209. 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, Springer-Verlag, New York, 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), 3521-3529. 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), 133-141. 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), 93-111.  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), 673-716. 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), 1699-1708. doi: 10.1016/j.na.2009.01.007.  Google Scholar

show all references

References:
[1]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 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), 181-209. 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, Springer-Verlag, New York, 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), 3521-3529. 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), 133-141. 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), 93-111.  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), 673-716. 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), 1699-1708. doi: 10.1016/j.na.2009.01.007.  Google Scholar

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