\omega-limit sets for porous medium equation with initial data in some weighted spaces

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January 2013, 18(1): 223-236. doi: 10.3934/dcdsb.2013.18.223

$\omega$-limit sets for porous medium equation with initial data in some weighted spaces

Liangwei Wang ^1, , Jingxue Yin ^2, and Chunhua Jin ^2,

1.	School of Math. Stat., Chongqing Three Gorges Univ., Wanzhou 404000, China
2.	School of Math. Sci., South China Normal Univ., Guangzhou 510631, China, China

Received May 2012 Revised August 2012 Published September 2012

Abstract
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We discuss the $\omega$-limit set for the Cauchy problem of the porous medium equation with initial data in some weighted spaces. Exactly, we show that there exists some relationship between the $\omega$-limit set of the rescaled initial data and the $\omega$-limit set of the spatially rescaled version of solutions. We also give some applications of such a relationship.

Keywords: weighted space., $\omega$-limit set, Porous medium equation.

Mathematics Subject Classification: 35K55, 35B4.

Citation: Liangwei Wang, Jingxue Yin, Chunhua Jin. $\omega$-limit sets for porous medium equation with initial data in some weighted spaces. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 223-236. doi: 10.3934/dcdsb.2013.18.223

References:

[1]	S. Kamenomostskaya, The asymptotic behaviour of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87.
[2]	Ph. Bénilan, "Opérateurs Accrétifs et Semi-Groupes dans les Espaces $L^p$ ($1\leq p \leq\infty$)," France-Japan Seminar, Tokyo, 1976.
[3]	L. Véron, Coercivité et propriétés régularisantes des semi-groupes non linéaires dans les espaces de Banach, Ann. Fac. Sci. Toulouse, 1 (1979), 171-200. doi: 10.5802/afst.535.
[4]	N. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. Math. J., 30 (1981), 749-785. doi: 10.1512/iumj.1981.30.30056.
[5]	S. Kamin and L. A. Peletier, Large time behaviour of solutions of the porous media equation with absorption, Israel J. Math., 55 (1986), 129-146.
[6]	F. Quirós and J. L. Vazquez, Asymptotic behaviour of the porous media equation in an exterior domain, Ann. Scuola Normale Sup. Pisa, 28 (1999), 183-227.
[7]	J. A. Carrillo and K. Fellner, Long-time asymptotics via entropy methods for diffusion dominated equations, Asymptotic Analysis, 42 (2005), 29-54.
[8]	G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493-508.
[9]	A. Friedman and S. Kamin, The asymptotic behavior of gas in an N-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563.
[10]	S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana, 4 (1988), 339-354.
[11]	J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118.
[12]	N. Alikakos and R. Rostamian, On the uniformization of the solutions of the porous medium equation in $\mathbbR^N$, IsraelJ. Math., 47 (1984), 270-290.
[13]	J. L. Vázquez and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data, Chin. Ann. Math. Ser. B, 23 (2002), 293-310.
[14]	T. Cazenave, F. Dickstein and F. B. Weissler, Universal solutions of the heat equation on $\mathbbR^N$, Discrete Contin. Dyn. Sys., 9 (2003), 1105-1132.
[15]	T. Cazenave, F. Dickstein and F. B. Weissler, Universal solutions of a nonlinear heat equation on $\mathbbR^N$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (2003), 77-117.
[16]	T. Cazenave, F. Dickstein and F. B. Weissler, Chaotic behavior of solutions of the Navier-Stokes system in $\mathbbR^N$, Adv. Differ. Equations, 10 (2005), 361-398.
[17]	J. A. Carrillo and J. L. Vázquez, Asymptotic complexity infiltration equations, J. Evol. Equ., 7 (2007), 471-495.
[18]	T. Cazenave, F. Dickstein and F. B. Weissler, Nonparabolic asymptotic limits of solutions of the heat equation on$\mathbbR^N$, J. Dyn. Diff. Eqns., 19 (2007), 789-818.
[19]	J. X. Yin, L. W. Wang and R. Huang, Complexity of asymptotic behavior of solutions for the porous medium equations, Acta Mathematica Scientia, 30 (2010), 1865-1880.
[20]	J. X. Yin, L. W. Wang and R. Huang, Complexity of a symptotic behavior of the porous medium equation in $\mathbbR^N$, J. Evol. Equ., 11 (2011), 429-455. doi: 10.1007/s00028-010-0097-4.
[21]	E. DiBenedetto, Continuity of weak solutions to ageneral porous media equation, Indiana Univ. Math. J., 32 (1983), 83-118.
[22]	P. Bénilan, M. G. Crandall and M. Pierre, Solutions of the porousmedium in $\mathbbR^N$ under optimal conditions on the initialvalues, Indiana Univ. Math. J., 33 (1984), 51-87.
[23]	E. DiBenedetto, "Degenerate Parabolic Equations," New York, Springer-Verlag, 1993.
[24]	J. L. Vázquez, "The Porous Medium Equation: MathematicalTheory, Oxford Mathematical Monographs," Oxford/New York, TheClarendon Press/Oxford University Press, 2008.
[25]	J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Parabolic Equations: Equations of Porous Medium Type," Oxford University Press, 2006.
[26]	L. A. Caffarelli, J. L. Vázquez and N. I. Wolanski, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J., 33 (1984), 51-87.
[27]	J. N. Zhao and H. J. Yuan, Lipschitz continuity of solutions and interfaces of the evolution $p$-Laplacian equation, Northeast. Math. J., 8 (1992), 21-37.
[28]	T. Cazenave, F. Dickstein, M. Escobedo and F. B. Weissler, Self-similar solutions of a nonlinear heat equation, J. Math. Sci. Univ. Tokyo, 8 (2001), 501-540.

show all references

References:

[1]	S. Kamenomostskaya, The asymptotic behaviour of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87.
[2]	Ph. Bénilan, "Opérateurs Accrétifs et Semi-Groupes dans les Espaces $L^p$ ($1\leq p \leq\infty$)," France-Japan Seminar, Tokyo, 1976.
[3]	L. Véron, Coercivité et propriétés régularisantes des semi-groupes non linéaires dans les espaces de Banach, Ann. Fac. Sci. Toulouse, 1 (1979), 171-200. doi: 10.5802/afst.535.
[4]	N. Alikakos and R. Rostamian, Large time behavior of solutions of Neumann boundary value problem for the porous medium equation, Indiana Univ. Math. J., 30 (1981), 749-785. doi: 10.1512/iumj.1981.30.30056.
[5]	S. Kamin and L. A. Peletier, Large time behaviour of solutions of the porous media equation with absorption, Israel J. Math., 55 (1986), 129-146.
[6]	F. Quirós and J. L. Vazquez, Asymptotic behaviour of the porous media equation in an exterior domain, Ann. Scuola Normale Sup. Pisa, 28 (1999), 183-227.
[7]	J. A. Carrillo and K. Fellner, Long-time asymptotics via entropy methods for diffusion dominated equations, Asymptotic Analysis, 42 (2005), 29-54.
[8]	G. Reyes and J. L. Vázquez, Long time behavior for the inhomogeneous PME in a medium with slowly decaying density, Commun. Pure Appl. Anal., 8 (2009), 493-508.
[9]	A. Friedman and S. Kamin, The asymptotic behavior of gas in an N-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551-563.
[10]	S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana, 4 (1988), 339-354.
[11]	J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118.
[12]	N. Alikakos and R. Rostamian, On the uniformization of the solutions of the porous medium equation in $\mathbbR^N$, IsraelJ. Math., 47 (1984), 270-290.
[13]	J. L. Vázquez and E. Zuazua, Complexity of large time behaviour of evolution equations with bounded data, Chin. Ann. Math. Ser. B, 23 (2002), 293-310.
[14]	T. Cazenave, F. Dickstein and F. B. Weissler, Universal solutions of the heat equation on $\mathbbR^N$, Discrete Contin. Dyn. Sys., 9 (2003), 1105-1132.
[15]	T. Cazenave, F. Dickstein and F. B. Weissler, Universal solutions of a nonlinear heat equation on $\mathbbR^N$, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 5 (2003), 77-117.
[16]	T. Cazenave, F. Dickstein and F. B. Weissler, Chaotic behavior of solutions of the Navier-Stokes system in $\mathbbR^N$, Adv. Differ. Equations, 10 (2005), 361-398.
[17]	J. A. Carrillo and J. L. Vázquez, Asymptotic complexity infiltration equations, J. Evol. Equ., 7 (2007), 471-495.
[18]	T. Cazenave, F. Dickstein and F. B. Weissler, Nonparabolic asymptotic limits of solutions of the heat equation on$\mathbbR^N$, J. Dyn. Diff. Eqns., 19 (2007), 789-818.
[19]	J. X. Yin, L. W. Wang and R. Huang, Complexity of asymptotic behavior of solutions for the porous medium equations, Acta Mathematica Scientia, 30 (2010), 1865-1880.
[20]	J. X. Yin, L. W. Wang and R. Huang, Complexity of a symptotic behavior of the porous medium equation in $\mathbbR^N$, J. Evol. Equ., 11 (2011), 429-455. doi: 10.1007/s00028-010-0097-4.
[21]	E. DiBenedetto, Continuity of weak solutions to ageneral porous media equation, Indiana Univ. Math. J., 32 (1983), 83-118.
[22]	P. Bénilan, M. G. Crandall and M. Pierre, Solutions of the porousmedium in $\mathbbR^N$ under optimal conditions on the initialvalues, Indiana Univ. Math. J., 33 (1984), 51-87.
[23]	E. DiBenedetto, "Degenerate Parabolic Equations," New York, Springer-Verlag, 1993.
[24]	J. L. Vázquez, "The Porous Medium Equation: MathematicalTheory, Oxford Mathematical Monographs," Oxford/New York, TheClarendon Press/Oxford University Press, 2008.
[25]	J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Parabolic Equations: Equations of Porous Medium Type," Oxford University Press, 2006.
[26]	L. A. Caffarelli, J. L. Vázquez and N. I. Wolanski, Lipschitz continuity of solutions and interfaces of the N-dimensional porous medium equation, Indiana Univ. Math. J., 33 (1984), 51-87.
[27]	J. N. Zhao and H. J. Yuan, Lipschitz continuity of solutions and interfaces of the evolution $p$-Laplacian equation, Northeast. Math. J., 8 (1992), 21-37.
[28]	T. Cazenave, F. Dickstein, M. Escobedo and F. B. Weissler, Self-similar solutions of a nonlinear heat equation, J. Math. Sci. Univ. Tokyo, 8 (2001), 501-540.

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