1-dimensional Harnack estimates

Previous Article
Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel
DCDS-S Home
This Issue
Next Article
A singular limit problem for the Ibragimov-Shabat equation

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

Abstract
Related Papers

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 and 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.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

show all references

References:

[1]	E. DiBenedetto, Degenerate Parabolic Equations, Universitext, Springer-Verlag, New York, 1993. doi: 10.1007/978-0-387-94020-5.
[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.
[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.
[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.
[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.
[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.
[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.
[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.

[1]	Simona Fornaro, Maria Sosio, Vincenzo Vespri. $L^r_{ loc}-L^\infty_{ loc}$ estimates and expansion of positivity for a class of doubly non linear singular parabolic equations. Discrete and Continuous Dynamical Systems - S, 2014, 7 (4) : 737-760. doi: 10.3934/dcdss.2014.7.737
[2]	L. Cherfils, Y. Il'yasov. On the stationary solutions of generalized reaction diffusion equations with $p\& q$-Laplacian. Communications on Pure and Applied Analysis, 2005, 4 (1) : 9-22. doi: 10.3934/cpaa.2005.4.9
[3]	Michele V. Bartuccelli, K. B. Blyuss, Y. N. Kyrychko. Length scales and positivity of solutions of a class of reaction-diffusion equations. Communications on Pure and Applied Analysis, 2004, 3 (1) : 25-40. doi: 10.3934/cpaa.2004.3.25
[4]	Maurizio Garrione, Marta Strani. Monotone wave fronts for $(p, q)$-Laplacian driven reaction-diffusion equations. Discrete and Continuous Dynamical Systems - S, 2019, 12 (1) : 91-103. doi: 10.3934/dcdss.2019006
[5]	Shanming Ji, Yutian Li, Rui Huang, Xuejing Yin. Singular periodic solutions for the p-laplacian ina punctured domain. Communications on Pure and Applied Analysis, 2017, 16 (2) : 373-392. doi: 10.3934/cpaa.2017019
[6]	Meiqiang Feng, Yichen Zhang. Positive solutions of singular multiparameter p-Laplacian elliptic systems. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 1121-1147. doi: 10.3934/dcdsb.2021083
[7]	T. F. Ma, M. L. Pelicer. Attractors for weakly damped beam equations with $p$-Laplacian. Conference Publications, 2013, 2013 (special) : 525-534. doi: 10.3934/proc.2013.2013.525
[8]	Nikolaos S. Papageorgiou, George Smyrlis. Positive solutions for parametric $p$-Laplacian equations. Communications on Pure and Applied Analysis, 2016, 15 (5) : 1545-1570. doi: 10.3934/cpaa.2016002
[9]	C. Fabry, Raul Manásevich. Equations with a $p$-Laplacian and an asymmetric nonlinear term. Discrete and Continuous Dynamical Systems, 2001, 7 (3) : 545-557. doi: 10.3934/dcds.2001.7.545
[10]	Naoufel Ben Abdallah, Antoine Mellet, Marjolaine Puel. Fractional diffusion limit for collisional kinetic equations: A Hilbert expansion approach. Kinetic and Related Models, 2011, 4 (4) : 873-900. doi: 10.3934/krm.2011.4.873
[11]	Patrizia Pucci, Mingqi Xiang, Binlin Zhang. A diffusion problem of Kirchhoff type involving the nonlocal fractional p-Laplacian. Discrete and Continuous Dynamical Systems, 2017, 37 (7) : 4035-4051. doi: 10.3934/dcds.2017171
[12]	Jochen Merker, Aleš Matas. Positivity of self-similar solutions of doubly nonlinear reaction-diffusion equations. Conference Publications, 2015, 2015 (special) : 817-825. doi: 10.3934/proc.2015.0817
[13]	Patrizia Pucci, Raffaella Servadei. Nonexistence for $p$--Laplace equations with singular weights. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1421-1438. doi: 10.3934/cpaa.2010.9.1421
[14]	Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Positive solutions for p-Laplacian equations with concave terms. Conference Publications, 2011, 2011 (Special) : 922-930. doi: 10.3934/proc.2011.2011.922
[15]	Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Existence of radial solutions for the $p$-Laplacian elliptic equations with weights. Discrete and Continuous Dynamical Systems, 2006, 15 (2) : 447-479. doi: 10.3934/dcds.2006.15.447
[16]	Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete and Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[17]	John R. Graef, Lingju Kong, Min Wang. Existence of homoclinic solutions for second order difference equations with $p$-laplacian. Conference Publications, 2015, 2015 (special) : 533-539. doi: 10.3934/proc.2015.0533
[18]	Elisa Calzolari, Roberta Filippucci, Patrizia Pucci. Dead cores and bursts for p-Laplacian elliptic equations with weights. Conference Publications, 2007, 2007 (Special) : 191-200. doi: 10.3934/proc.2007.2007.191
[19]	Zuodong Yang, Jing Mo, Subei Li. Positive solutions of $p$-Laplacian equations with nonlinear boundary condition. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 623-636. doi: 10.3934/dcdsb.2011.16.623
[20]	Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1031-1051. doi: 10.3934/cpaa.2009.8.1031

2020 Impact Factor: 2.425

Tools

Metrics

Other articles
by authors

on AIMS
Fatma Gamze Düzgün Ugo Gianazza Vincenzo Vespri
on Google Scholar
Fatma Gamze Düzgün Ugo Gianazza Vincenzo Vespri

[Back to Top]