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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

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 & 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.  Google Scholar

[2]

Ph. Bénilan, "Opérateurs Accrétifs et Semi-Groupes dans les Espaces $L^p$ ($1\leq p \leq\infty$)," France-Japan Seminar, Tokyo, 1976. Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. A. Carrillo and K. Fellner, Long-time asymptotics via entropy methods for diffusion dominated equations, Asymptotic Analysis, 42 (2005), 29-54.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana, 4 (1988), 339-354.  Google Scholar

[11]

J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. A. Carrillo and J. L. Vázquez, Asymptotic complexity infiltration equations, J. Evol. Equ., 7 (2007), 471-495.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. DiBenedetto, Continuity of weak solutions to ageneral porous media equation, Indiana Univ. Math. J., 32 (1983), 83-118.  Google Scholar

[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.  Google Scholar

[23]

E. DiBenedetto, "Degenerate Parabolic Equations," New York, Springer-Verlag, 1993.  Google Scholar

[24]

J. L. Vázquez, "The Porous Medium Equation: MathematicalTheory, Oxford Mathematical Monographs," Oxford/New York, TheClarendon Press/Oxford University Press, 2008.  Google Scholar

[25]

J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Parabolic Equations: Equations of Porous Medium Type," Oxford University Press, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

S. Kamenomostskaya, The asymptotic behaviour of the solution of the filtration equation, Israel J. Math., 14 (1973), 76-87.  Google Scholar

[2]

Ph. Bénilan, "Opérateurs Accrétifs et Semi-Groupes dans les Espaces $L^p$ ($1\leq p \leq\infty$)," France-Japan Seminar, Tokyo, 1976. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

J. A. Carrillo and K. Fellner, Long-time asymptotics via entropy methods for diffusion dominated equations, Asymptotic Analysis, 42 (2005), 29-54.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Kamin and J. L. Vázquez, Fundamental solutions and asymptotic behaviour for the p-Laplacian equation, Rev. Mat. Iberoamericana, 4 (1988), 339-354.  Google Scholar

[11]

J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space, J. Evol. Equ., 3 (2003), 67-118.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

J. A. Carrillo and J. L. Vázquez, Asymptotic complexity infiltration equations, J. Evol. Equ., 7 (2007), 471-495.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[21]

E. DiBenedetto, Continuity of weak solutions to ageneral porous media equation, Indiana Univ. Math. J., 32 (1983), 83-118.  Google Scholar

[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.  Google Scholar

[23]

E. DiBenedetto, "Degenerate Parabolic Equations," New York, Springer-Verlag, 1993.  Google Scholar

[24]

J. L. Vázquez, "The Porous Medium Equation: MathematicalTheory, Oxford Mathematical Monographs," Oxford/New York, TheClarendon Press/Oxford University Press, 2008.  Google Scholar

[25]

J. L. Vázquez, "Smoothing and Decay Estimates for Nonlinear Parabolic Equations: Equations of Porous Medium Type," Oxford University Press, 2006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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