Stability of degenerate parabolic Cauchy problems

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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

Abstract
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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

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