# American Institute of Mathematical Sciences

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

 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.
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:
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##### References:
 [1] S. Kamenomostskaya, The asymptotic behaviour of the solution of the filtration equation,, Israel J. Math., 14 (1973), 76.   Google Scholar [2] Ph. Bénilan, "Opérateurs Accrétifs et Semi-Groupes dans les Espaces $L^p$ ($1\leq p \leq\infty$),", France-Japan Seminar, (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.  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.  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.   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.   Google Scholar [7] J. A. Carrillo and K. Fellner, Long-time asymptotics via entropy methods for diffusion dominated equations,, Asymptotic Analysis, 42 (2005), 29.   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.   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.   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.   Google Scholar [11] J. L. Vázquez, Asymptotic behaviour for the porous medium equation posed in the whole space,, J. Evol. Equ., 3 (2003), 67.   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.   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.   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.   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.   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.   Google Scholar [17] J. A. Carrillo and J. L. Vázquez, Asymptotic complexity infiltration equations,, J. Evol. Equ., 7 (2007), 471.   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.   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.   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.  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.   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.   Google Scholar [23] E. DiBenedetto, "Degenerate Parabolic Equations,", New York, (1993).   Google Scholar [24] J. L. Vázquez, "The Porous Medium Equation: MathematicalTheory, Oxford Mathematical Monographs,", Oxford/New York, (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.   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.   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.   Google Scholar
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