August  2013, 6(4): 1029-1042. doi: 10.3934/dcdss.2013.6.1029

Pointwise estimates for solutions of singular quasi-linear parabolic equations

1. 

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

2. 

Institute of Applied Mathematics and Mechanics, Donetsk 83114, Ukraine

Received  March 2011 Revised  September 2011 Published  December 2012

For a class of singular divergence type quasi-linear parabolicequations with a Radon measure on the right hand side we derivepointwise estimates for solutions via the nonlinear Wolffpotentials.
Citation: 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
References:
[1]

Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (III), 125 (1957), 25-43.  Google Scholar

[2]

Springer, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[3]

in "Handbook of Differential Equations. Evolution Equations" (eds. C. Dafermos and E. Feireisl), Elsevier, 1 (2004), 169-286.  Google Scholar

[4]

Acta Mathematica, 200 (2008), 181-209. doi: 10.1007/s11511-008-0026-3.  Google Scholar

[5]

Amer. J. Math., to appear. doi: 10.1353/ajm.2011.0023.  Google Scholar

[6]

Lecture Notes in Math., 481, Springer, 1975.  Google Scholar

[7]

Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[8]

Duke Math. J., 111 (2002), 1-49. doi: 10.1215/S0012-7094-02-11111-9.  Google Scholar

[9]

Translations of Mathematical Monographs, 23 American Mathematical Society, Providence, R.I. 1967.  Google Scholar

[10]

preprint 2010.  Google Scholar

[11]

J. Diff. Eq., 247 (2009), 2740-2777. doi: 10.1016/j.jde.2009.08.018.  Google Scholar

[12]

Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence, RI, 1997.  Google Scholar

[13]

Ann. of Math. (2), 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859.  Google Scholar

[14]

J. Funct. Anal., 256 (2009), 1875-1906. doi: 10.1016/j.jfa.2009.01.012.  Google Scholar

[15]

Dokl. Akad. Nauk, 398 (2004), 458-461. (Russian)  Google Scholar

[16]

Amer. J. Math.,124 (2002), 369-410.  Google Scholar

show all references

References:
[1]

Mem. Accad. Sci. Torino Cl. Sci. Fis. Mat. Nat. (III), 125 (1957), 25-43.  Google Scholar

[2]

Springer, New York, 1993. doi: 10.1007/978-1-4612-0895-2.  Google Scholar

[3]

in "Handbook of Differential Equations. Evolution Equations" (eds. C. Dafermos and E. Feireisl), Elsevier, 1 (2004), 169-286.  Google Scholar

[4]

Acta Mathematica, 200 (2008), 181-209. doi: 10.1007/s11511-008-0026-3.  Google Scholar

[5]

Amer. J. Math., to appear. doi: 10.1353/ajm.2011.0023.  Google Scholar

[6]

Lecture Notes in Math., 481, Springer, 1975.  Google Scholar

[7]

Acta Math., 172 (1994), 137-161. doi: 10.1007/BF02392793.  Google Scholar

[8]

Duke Math. J., 111 (2002), 1-49. doi: 10.1215/S0012-7094-02-11111-9.  Google Scholar

[9]

Translations of Mathematical Monographs, 23 American Mathematical Society, Providence, R.I. 1967.  Google Scholar

[10]

preprint 2010.  Google Scholar

[11]

J. Diff. Eq., 247 (2009), 2740-2777. doi: 10.1016/j.jde.2009.08.018.  Google Scholar

[12]

Mathematical Surveys and Monographs, 51. American Mathematical Society, Providence, RI, 1997.  Google Scholar

[13]

Ann. of Math. (2), 168 (2008), 859-914. doi: 10.4007/annals.2008.168.859.  Google Scholar

[14]

J. Funct. Anal., 256 (2009), 1875-1906. doi: 10.1016/j.jfa.2009.01.012.  Google Scholar

[15]

Dokl. Akad. Nauk, 398 (2004), 458-461. (Russian)  Google Scholar

[16]

Amer. J. Math.,124 (2002), 369-410.  Google Scholar

[1]

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

[2]

Gary M. Lieberman. Schauder estimates for singular parabolic and elliptic equations of Keldysh type. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1525-1566. doi: 10.3934/dcdsb.2016010

[3]

Farman Mamedov, Sara Monsurrò, Maria Transirico. Potential estimates and applications to elliptic equations. Conference Publications, 2015, 2015 (special) : 793-800. doi: 10.3934/proc.2015.0793

[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]

Marina Ghisi, Massimo Gobbino. Hyperbolic--parabolic singular perturbation for mildly degenerate Kirchhoff equations: Global-in-time error estimates. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1313-1332. doi: 10.3934/cpaa.2009.8.1313

[6]

Minh-Phuong Tran, Thanh-Nhan Nguyen. Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete & Continuous Dynamical Systems, 2021, 41 (9) : 4461-4476. doi: 10.3934/dcds.2021043

[7]

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 & Continuous Dynamical Systems - S, 2014, 7 (4) : 737-760. doi: 10.3934/dcdss.2014.7.737

[8]

Wu Chen, Zhongxue Lu. Existence and nonexistence of positive solutions to an integral system involving Wolff potential. Communications on Pure & Applied Analysis, 2016, 15 (2) : 385-398. doi: 10.3934/cpaa.2016.15.385

[9]

Huan Chen, Zhongxue Lü. The properties of positive solutions to an integral system involving Wolff potential. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1879-1904. doi: 10.3934/dcds.2014.34.1879

[10]

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

[11]

Zhigang Wu, Weike Wang. Pointwise estimates of solutions for the Euler-Poisson equations with damping in multi-dimensions. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 1101-1117. doi: 10.3934/dcds.2010.26.1101

[12]

Luisa Moschini, Guillermo Reyes, Alberto Tesei. Nonuniqueness of solutions to semilinear parabolic equations with singular coefficients. Communications on Pure & Applied Analysis, 2006, 5 (1) : 155-179. doi: 10.3934/cpaa.2006.5.155

[13]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605

[14]

Chiun-Chuan Chen, Chang-Shou Lin. Mean field equations of Liouville type with singular data: Sharper estimates. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1237-1272. doi: 10.3934/dcds.2010.28.1237

[15]

Yoshikazu Giga, Robert V. Kohn. Scale-invariant extinction time estimates for some singular diffusion equations. Discrete & Continuous Dynamical Systems, 2011, 30 (2) : 509-535. doi: 10.3934/dcds.2011.30.509

[16]

Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 761-790. doi: 10.3934/dcdss.2011.4.761

[17]

Huijiang Zhao. Large time decay estimates of solutions of nonlinear parabolic equations. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 69-114. doi: 10.3934/dcds.2002.8.69

[18]

Emmanuele DiBenedetto, Ugo Gianazza and Vincenzo Vespri. Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations. Electronic Research Announcements, 2006, 12: 95-99.

[19]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977

[20]

Ping Li, Pablo Raúl Stinga, José L. Torrea. On weighted mixed-norm Sobolev estimates for some basic parabolic equations. Communications on Pure & Applied Analysis, 2017, 16 (3) : 855-882. doi: 10.3934/cpaa.2017041

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (33)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]