American Institute of Mathematical Sciences

Advanced
January  2015, 14(1): 201-216. doi: 10.3934/cpaa.2015.14.201

Stability of degenerate parabolic Cauchy problems

Teemu Lukkari 1, and  Mikko Parviainen 1,

1. 

Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of Jyväskylä, Finland, Finland

Received  January 2014 Revised  February 2014 Published  September 2014

We prove that solutions to Cauchy problems related to the $p$-parabolic equations are stable with respect to the nonlinearity exponent $p$. More specifically, solutions with a fixed initial trace converge in an $L^q$-space to a solution of the limit problem as $p>2$ varies.
Keywords: Cauchy problems, stability, initial value problems, Barenblatt solutions., Nonlinear parabolic equations.
Mathematics Subject Classification: Primary: 35K55; Secondary: 35K15, 35K6.
Citation: Teemu Lukkari, Mikko Parviainen. Stability of degenerate parabolic Cauchy problems. Communications on Pure & Applied Analysis, 2015, 14 (1) : 201-216. doi: 10.3934/cpaa.2015.14.201
References:
[1]

L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data,, J. Funct. Anal., 147 (1997), 237. doi: 10.1006/jfan.1996.3040. Google Scholar

[2]

E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation,, Trans. Amer. Math. Soc., 314 (1989), 187. doi: 10.2307/2001442. Google Scholar

[3]

E. DiBenedetto, Degenerate Parabolic Equations,, Universitext. Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2. Google Scholar

[4]

Y. Fujishima, J. Habermann, J. Kinnunen and M. Masson, Stability for parabolic quasiminimizers,, Potential Anal. (to appear). Available at , (). Google Scholar

[5]

S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the $p$-Laplacian equation,, Rev. Mat. Iberoam., 4 (1988), 339. doi: 10.4171/RMI/77. Google Scholar

[6]

J. Kinnunen and J. Lewis, Higher integrability for parabolic systems of $p$-Laplacian type,, Duke Math J., 102 (2000), 253. doi: 10.1215/S0012-7094-00-10223-2. Google Scholar

[7]

J. Kinnunen and P. Lindqvist, Summability of semicontinuous supersolutions to a quasilinear parabolic equation,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4 (2005), 59. Google Scholar

[8]

J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation,, Ann. Mat. Pura Appl., 185 (2006), 411. doi: 10.1007/s10231-005-0160-x. Google Scholar

[9]

J. Kinnunen and M. Parviainen, Stability for degenerate parabolic equations,, Adv. Calc. Var., 3 (2010), 29. doi: 10.1515/ACV.2010.002. Google Scholar

[10]

T. Kuusi and M. Parviainen, Existence for a degenerate Cauchy problem,, Manuscripta Math., 128 (2009), 213. doi: 10.1007/s00229-008-0232-5. Google Scholar

[11]

G. Li and O. Martio, Stability of solutions of varying degenerate elliptic equations,, Indiana Univ. Math. J., 47 (1998), 873. doi: 10.1512/iumj.1998.47.1458. Google Scholar

[12]

P. Lindqvist, Stability for the solutions of div$(|\nabla u |^{p-2}\nabla u)=f$ with varying $p$,, J. Math. Anal. Appl., 127 (1987), 93. doi: 10.1016/0022-247X(87)90142-9. Google Scholar

[13]

P. Lindqvist, On nonlinear Rayleigh quotients,, Potential Anal., 2 (1993), 199. doi: 10.1007/BF01048505. Google Scholar

[14]

T. Lukkari, Stability of solutions to nonlinear diffusion equations,, Submitted. Available at , (). Google Scholar

[15]

J. Naumann, Einführung in die Theorie parabolischer Variationsungleichungen, volume 64 of Teubner-Texte zur Mathematik,, BSB B. G. Teubner Verlagsgesellschaft, (1984). Google Scholar

[16]

R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations,, volume 49 of Mathematical Surveys and Monographs, (1997). Google Scholar

[17]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[18]

J. L. Vázquez, The Porous Medium Equation-Mathematical Theory,, Oxford University Press, (2007). Google Scholar

show all references

References:
[1]

L. Boccardo, A. Dall'Aglio, T. Gallouët and L. Orsina, Nonlinear parabolic equations with measure data,, J. Funct. Anal., 147 (1997), 237. doi: 10.1006/jfan.1996.3040. Google Scholar

[2]

E. DiBenedetto and M. A. Herrero, On the Cauchy problem and initial traces for a degenerate parabolic equation,, Trans. Amer. Math. Soc., 314 (1989), 187. doi: 10.2307/2001442. Google Scholar

[3]

E. DiBenedetto, Degenerate Parabolic Equations,, Universitext. Springer-Verlag, (1993). doi: 10.1007/978-1-4612-0895-2. Google Scholar

[4]

Y. Fujishima, J. Habermann, J. Kinnunen and M. Masson, Stability for parabolic quasiminimizers,, Potential Anal. (to appear). Available at , (). Google Scholar

[5]

S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the $p$-Laplacian equation,, Rev. Mat. Iberoam., 4 (1988), 339. doi: 10.4171/RMI/77. Google Scholar

[6]

J. Kinnunen and J. Lewis, Higher integrability for parabolic systems of $p$-Laplacian type,, Duke Math J., 102 (2000), 253. doi: 10.1215/S0012-7094-00-10223-2. Google Scholar

[7]

J. Kinnunen and P. Lindqvist, Summability of semicontinuous supersolutions to a quasilinear parabolic equation,, Ann. Sc. Norm. Super. Pisa Cl. Sci., 4 (2005), 59. Google Scholar

[8]

J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation,, Ann. Mat. Pura Appl., 185 (2006), 411. doi: 10.1007/s10231-005-0160-x. Google Scholar

[9]

J. Kinnunen and M. Parviainen, Stability for degenerate parabolic equations,, Adv. Calc. Var., 3 (2010), 29. doi: 10.1515/ACV.2010.002. Google Scholar

[10]

T. Kuusi and M. Parviainen, Existence for a degenerate Cauchy problem,, Manuscripta Math., 128 (2009), 213. doi: 10.1007/s00229-008-0232-5. Google Scholar

[11]

G. Li and O. Martio, Stability of solutions of varying degenerate elliptic equations,, Indiana Univ. Math. J., 47 (1998), 873. doi: 10.1512/iumj.1998.47.1458. Google Scholar

[12]

P. Lindqvist, Stability for the solutions of div$(|\nabla u |^{p-2}\nabla u)=f$ with varying $p$,, J. Math. Anal. Appl., 127 (1987), 93. doi: 10.1016/0022-247X(87)90142-9. Google Scholar

[13]

P. Lindqvist, On nonlinear Rayleigh quotients,, Potential Anal., 2 (1993), 199. doi: 10.1007/BF01048505. Google Scholar

[14]

T. Lukkari, Stability of solutions to nonlinear diffusion equations,, Submitted. Available at , (). Google Scholar

[15]

J. Naumann, Einführung in die Theorie parabolischer Variationsungleichungen, volume 64 of Teubner-Texte zur Mathematik,, BSB B. G. Teubner Verlagsgesellschaft, (1984). Google Scholar

[16]

R. E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations,, volume 49 of Mathematical Surveys and Monographs, (1997). Google Scholar

[17]

J. Simon, Compact sets in the space $L^p(0,T;B)$,, Ann. Mat. Pura Appl., 146 (1987), 65. doi: 10.1007/BF01762360. Google Scholar

[18]

J. L. Vázquez, The Porous Medium Equation-Mathematical Theory,, Oxford University Press, (2007). Google Scholar

[1]

John V. Baxley, Philip T. Carroll. Nonlinear boundary value problems with multiple positive solutions. Conference Publications, 2003, 2003 (Special) : 83-90. doi: 10.3934/proc.2003.2003.83
[2]

Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059
[3]

Leo G. Rebholz, Dehua Wang, Zhian Wang, Camille Zerfas, Kun Zhao. Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3789-3838. doi: 10.3934/dcds.2019154
[4]

Sergei Avdonin, Fritz Gesztesy, Konstantin A. Makarov. Spectral estimation and inverse initial boundary value problems. Inverse Problems & Imaging, 2010, 4 (1) : 1-9. doi: 10.3934/ipi.2010.4.1
[5]

Hiroshi Watanabe. Solvability of boundary value problems for strongly degenerate parabolic equations with discontinuous coefficients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (1) : 177-189. doi: 10.3934/dcdss.2014.7.177
[6]

Sabri Bensid, Jesús Ildefonso Díaz. Stability results for discontinuous nonlinear elliptic and parabolic problems with a S-shaped bifurcation branch of stationary solutions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1757-1778. doi: 10.3934/dcdsb.2017105
[7]

Runzhang Xu, Mingyou Zhang, Shaohua Chen, Yanbing Yang, Jihong Shen. The initial-boundary value problems for a class of sixth order nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5631-5649. doi: 10.3934/dcds.2017244
[8]

Grey Ballard, John Baxley, Nisrine Libbus. Qualitative behavior and computation of multiple solutions of nonlinear boundary value problems. Communications on Pure & Applied Analysis, 2006, 5 (2) : 251-259. doi: 10.3934/cpaa.2006.5.251
[9]

Inara Yermachenko, Felix Sadyrbaev. Types of solutions and multiplicity results for second order nonlinear boundary value problems. Conference Publications, 2007, 2007 (Special) : 1061-1069. doi: 10.3934/proc.2007.2007.1061
[10]

Ben-Yu Guo, Zhong-Qing Wang. A spectral collocation method for solving initial value problems of first order ordinary differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (3) : 1029-1054. doi: 10.3934/dcdsb.2010.14.1029
[11]

Hermen Jan Hupkes, Emmanuelle Augeraud-Véron. Well-posedness of initial value problems for functional differential and algebraic equations of mixed type. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 737-765. doi: 10.3934/dcds.2011.30.737
[12]

Aimin Huang, Roger Temam. The linear hyperbolic initial and boundary value problems in a domain with corners. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1627-1665. doi: 10.3934/dcdsb.2014.19.1627
[13]

Jin Liang, James H. Liu, Ti-Jun Xiao. Nonlocal Cauchy problems for nonautonomous evolution equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 529-535. doi: 10.3934/cpaa.2006.5.529
[14]

Zhiyuan Li, Xinchi Huang, Masahiro Yamamoto. Initial-boundary value problems for multi-term time-fractional diffusion equations with $ x $-dependent coefficients. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020001
[15]

Kenneth Hvistendahl Karlsen, Nils Henrik Risebro. On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1081-1104. doi: 10.3934/dcds.2003.9.1081
[16]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760
[17]

Feliz Minhós, Rui Carapinha. On higher order nonlinear impulsive boundary value problems. Conference Publications, 2015, 2015 (special) : 851-860. doi: 10.3934/proc.2015.0851
[18]

Jan Prüss, Gieri Simonett, Rico Zacher. On normal stability for nonlinear parabolic equations. Conference Publications, 2009, 2009 (Special) : 612-621. doi: 10.3934/proc.2009.2009.612
[19]

G. Infante. Positive solutions of nonlocal boundary value problems with singularities. Conference Publications, 2009, 2009 (Special) : 377-384. doi: 10.3934/proc.2009.2009.377
[20]

John R. Graef, Lingju Kong, Qingkai Kong, Min Wang. Positive solutions of nonlocal fractional boundary value problems. Conference Publications, 2013, 2013 (special) : 283-290. doi: 10.3934/proc.2013.2013.283

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences