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Time dependent perturbation in a non-autonomous non-classical parabolic equation
$\omega$-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 |
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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