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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), 237-258. 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), 187-224. doi: 10.2307/2001442.  Google Scholar

[3]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext. Springer-Verlag, New York, 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), 339-354. 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), 253-272. 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), 59-78.  Google Scholar

[8]

J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl., 185 (2006), 411-435. 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), 29-48. doi: 10.1515/ACV.2010.002.  Google Scholar

[10]

T. Kuusi and M. Parviainen, Existence for a degenerate Cauchy problem, Manuscripta Math., 128 (2009), 213-249. 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), 873-891. 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), 93-102. doi: 10.1016/0022-247X(87)90142-9.  Google Scholar

[13]

P. Lindqvist, On nonlinear Rayleigh quotients, Potential Anal., 2 (1993), 199-218. 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, Leipzig, 1984.  Google Scholar

[16]

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

[17]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[18]

J. L. Vázquez, The Porous Medium Equation-Mathematical Theory, Oxford University Press, Oxford, 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), 237-258. 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), 187-224. doi: 10.2307/2001442.  Google Scholar

[3]

E. DiBenedetto, Degenerate Parabolic Equations, Universitext. Springer-Verlag, New York, 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), 339-354. 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), 253-272. 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), 59-78.  Google Scholar

[8]

J. Kinnunen and P. Lindqvist, Pointwise behaviour of semicontinuous supersolutions to a quasilinear parabolic equation, Ann. Mat. Pura Appl., 185 (2006), 411-435. 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), 29-48. doi: 10.1515/ACV.2010.002.  Google Scholar

[10]

T. Kuusi and M. Parviainen, Existence for a degenerate Cauchy problem, Manuscripta Math., 128 (2009), 213-249. 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), 873-891. 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), 93-102. doi: 10.1016/0022-247X(87)90142-9.  Google Scholar

[13]

P. Lindqvist, On nonlinear Rayleigh quotients, Potential Anal., 2 (1993), 199-218. 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, Leipzig, 1984.  Google Scholar

[16]

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

[17]

J. Simon, Compact sets in the space $L^p(0,T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96. doi: 10.1007/BF01762360.  Google Scholar

[18]

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

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