American Institute of Mathematical Sciences

Advanced
July  2013, 12(4): 1731-1744. doi: 10.3934/cpaa.2013.12.1731

Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials

Vitali Liskevich 1, , Igor I. Skrypnik 2, and  Zeev Sobol 1,

1. 

Department of Mathematics, Swansea University, Swansea SA2 8PP, United Kingdom, United Kingdom
2. 

Institute of Applied Mathematics and Mechanics, Donetsk 83114, Ukraine

Received  April 2011 Revised  March 2012 Published  November 2012

For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.
Keywords: measure as a source, Parabolic $p$-Laplace equation, pointwise estimates, nonlinear Wolf potential..
Mathematics Subject Classification: Primary: 35K55, 35K65; Secondary: 35B4.
Citation: Vitali Liskevich, Igor I. Skrypnik, Zeev Sobol. Estimates of solutions for the parabolic $p$-Laplacian equation with measure via parabolic nonlinear potentials. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1731-1744. doi: 10.3934/cpaa.2013.12.1731
References:
[1]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (III), 125 (1957), 25.   Google Scholar

[2]

E. DiBenedetto, "Degenerate Parabolic Equations,", Springer, (1993).   Google Scholar

[3]

E. DiBenedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients,, Ann. Scuola Norm. Sup. Pisa Cl. Sci, 13 (1986), 487.   Google Scholar

[4]

E. DiBenedetto, U. Gianazza and V. Vespri, A Harnack inequality for a degenerate parabolic equation,, Acta Mathematica, 200 (2008), 181.   Google Scholar

[5]

F. Duzaar and G. Mingione, Gradient estimates in non-linear potential theory,, Rend. Lincei - Mat. Appl., 20 (2009), 179.   Google Scholar

[6]

F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials,, Amer. J. Math., 133 (2011), 1093.   Google Scholar

[7]

M. de Guzmán, A covering lemma with applications to differentiability of measures and singular integral operators,, Studia Math., 34 (1970), 299.   Google Scholar

[8]

M. de Guzmán, "Differentiation of Integrals in $R^n$,", Lecture Notes in Math., 481 (1975).   Google Scholar

[9]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137.   Google Scholar

[10]

D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.   Google Scholar

[11]

V. Liskevich and I. I. Skrypnik, Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes,, J. Diff. Eq., 247 (2009), 2740.   Google Scholar

[12]

V. Liskevich, I. I. Skrypnik and Z. Sobol, Potential estimates for quasi-linear parabolic equations,, Advanced Nonlinear Studies, 11 (2011), 905.   Google Scholar

[13]

J. Malý and W. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations,", Mathematical Surveys and Monographs, 51 ().   Google Scholar

[14]

N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859.   Google Scholar

[15]

N. C. Phuc and I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities,, J. Funct. Anal., 256 (2009), 1875.   Google Scholar

[16]

I. I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations (Russian),, Dokl. Akad. Nauk, 398 (2004), 458.   Google Scholar

[17]

N. Trudinger and X.-J. Wang, On the weak continuity of elliptic operators and applications to potential theory,, Amer. J. Math., 124 (2002), 369.   Google Scholar

show all references

References:
[1]

E. De Giorgi, Sulla differenziabilità e l'analiticità delle estremali degli integrali multipli regolari,, Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (III), 125 (1957), 25.   Google Scholar

[2]

E. DiBenedetto, "Degenerate Parabolic Equations,", Springer, (1993).   Google Scholar

[3]

E. DiBenedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients,, Ann. Scuola Norm. Sup. Pisa Cl. Sci, 13 (1986), 487.   Google Scholar

[4]

E. DiBenedetto, U. Gianazza and V. Vespri, A Harnack inequality for a degenerate parabolic equation,, Acta Mathematica, 200 (2008), 181.   Google Scholar

[5]

F. Duzaar and G. Mingione, Gradient estimates in non-linear potential theory,, Rend. Lincei - Mat. Appl., 20 (2009), 179.   Google Scholar

[6]

F. Duzaar and G. Mingione, Gradient estimates via non-linear potentials,, Amer. J. Math., 133 (2011), 1093.   Google Scholar

[7]

M. de Guzmán, A covering lemma with applications to differentiability of measures and singular integral operators,, Studia Math., 34 (1970), 299.   Google Scholar

[8]

M. de Guzmán, "Differentiation of Integrals in $R^n$,", Lecture Notes in Math., 481 (1975).   Google Scholar

[9]

T. Kilpeläinen and J. Malý, The Wiener test and potential estimates for quasilinear elliptic equations,, Acta Math., 172 (1994), 137.   Google Scholar

[10]

D. Labutin, Potential estimates for a class of fully nonlinear elliptic equations,, Duke Math. J., 111 (2002), 1.   Google Scholar

[11]

V. Liskevich and I. I. Skrypnik, Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes,, J. Diff. Eq., 247 (2009), 2740.   Google Scholar

[12]

V. Liskevich, I. I. Skrypnik and Z. Sobol, Potential estimates for quasi-linear parabolic equations,, Advanced Nonlinear Studies, 11 (2011), 905.   Google Scholar

[13]

J. Malý and W. Ziemer, "Fine Regularity of Solutions of Elliptic Partial Differential Equations,", Mathematical Surveys and Monographs, 51 ().   Google Scholar

[14]

N. C. Phuc and I. E. Verbitsky, Quasilinear and Hessian equations of Lane-Emden type,, Ann. of Math., 168 (2008), 859.   Google Scholar

[15]

N. C. Phuc and I. E. Verbitsky, Singular quasilinear and Hessian equations and inequalities,, J. Funct. Anal., 256 (2009), 1875.   Google Scholar

[16]

I. I. Skrypnik, On the Wiener criterion for quasilinear degenerate parabolic equations (Russian),, Dokl. Akad. Nauk, 398 (2004), 458.   Google Scholar

[17]

N. Trudinger and X.-J. Wang, On the weak continuity of elliptic operators and applications to potential theory,, Amer. J. Math., 124 (2002), 369.   Google Scholar

[1]

Chao Zhang, Xia Zhang, Shulin Zhou. Gradient estimates for the strong $p(x)$-Laplace equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4109-4129. doi: 10.3934/dcds.2017175
[2]

Margaret Beck. Stability of nonlinear waves: Pointwise estimates. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 191-211. doi: 10.3934/dcdss.2017010
[3]

Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427
[4]

Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51
[5]

Jong-Shenq Guo, Bei Hu. Blowup rate estimates for the heat equation with a nonlinear gradient source term. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 927-937. doi: 10.3934/dcds.2008.20.927
[6]

Chao Zhang, Lihe Wang, Shulin Zhou, Yun-Ho Kim. Global gradient estimates for $p(x)$-Laplace equation in non-smooth domains. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2559-2587. doi: 10.3934/cpaa.2014.13.2559
[7]

Vitali Liskevich, Igor I. Skrypnik. Pointwise estimates for solutions of singular quasi-linear parabolic equations. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 1029-1042. doi: 10.3934/dcdss.2013.6.1029
[8]

Peter I. Kogut, Olha P. Kupenko. On optimal control problem for an ill-posed strongly nonlinear elliptic equation with $p$-Laplace operator and $L^1$-type of nonlinearity. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1273-1295. doi: 10.3934/dcdsb.2019016
[9]

Yongqin Liu, Weike Wang. The pointwise estimates of solutions for dissipative wave equation in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1013-1028. doi: 10.3934/dcds.2008.20.1013
[10]

J.-P. Raymond. Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints. Discrete & Continuous Dynamical Systems - A, 1997, 3 (3) : 341-370. doi: 10.3934/dcds.1997.3.341
[11]

Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090
[12]

David Gómez-Castro, Juan Luis Vázquez. The fractional Schrödinger equation with singular potential and measure data. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7113-7139. doi: 10.3934/dcds.2019298
[13]

Arrigo Cellina. The regularity of solutions to some variational problems, including the p-Laplace equation for 3≤p < 4. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4071-4085. doi: 10.3934/dcds.2018177
[14]

Mikhail D. Surnachev, Vasily V. Zhikov. On existence and uniqueness classes for the Cauchy problem for parabolic equations of the p-Laplace type. Communications on Pure & Applied Analysis, 2013, 12 (4) : 1783-1812. doi: 10.3934/cpaa.2013.12.1783
[15]

N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073
[16]

Tian Xiang. Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion. Communications on Pure & Applied Analysis, 2019, 18 (1) : 255-284. doi: 10.3934/cpaa.2019014
[17]

Miao Liu, Weike Wang. Global existence and pointwise estimates of solutions for the multidimensional generalized Boussinesq-type equation. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1203-1222. doi: 10.3934/cpaa.2014.13.1203
[18]

Yutian Lei. Wolff type potential estimates and application to nonlinear equations with negative exponents. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 2067-2078. doi: 10.3934/dcds.2015.35.2067
[19]

Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1285-1301. doi: 10.3934/cpaa.2012.11.1285
[20]

Zhengce Zhang, Yanyan Li. Gradient blowup solutions of a semilinear parabolic equation with exponential source. Communications on Pure & Applied Analysis, 2013, 12 (1) : 269-280. doi: 10.3934/cpaa.2013.12.269

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences