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Sturm global attractors for $S^1$-equivariant parabolic equations
Grow up and slow decay in the critical Sobolev case
1. | Department of Applied Mathematics and Statistics, Comenius University, 84248 Bratislava, Slovak Republic |
2. | Division of Theoretical Mechanics, University of Nottingham, Nottingham NG7 2RD, United Kingdom |
References:
[1] |
M. Fila, J. R. King, M. Winkler and E. Yanagida, Optimal lower bound of the grow-up rate for a supercritical parabolic equation,, J. Differ. Equations, 228 (2006), 339.
doi: 10.1016/j.jde.2006.01.019. |
[2] |
M. Fila, J. R. King, M. Winkler and E. Yanagida, Grow-up rate of solutions of a semilinear parabolic equation with a critical exponent,, Adv. Differ. Equations, 12 (2007), 1.
|
[3] |
M. Fila, J. R. King, M. Winkler and E. Yanagida, Very slow grow-up of solutions of a semi-linear parabolic equation,, Proc. Edinb. Math. Soc., 53 (2011), 1.
doi: 10.1017/S0013091509001497. |
[4] |
M. Fila, H. Matano and P. Poláčik, Immediate regularization after blow-up,, SIAM J. Math. Anal., 37 (2005), 752.
doi: 10.1137/040613299. |
[5] |
M. Fila and M. Winkler, Rate of convergence to a singular steady state of a supercritical parabolic equation,, J. Evol. Equations, 8 (2008), 673.
doi: 10.1007/s00028-008-0400-9. |
[6] |
M. Fila, M. Winkler and E. Yanagida, Grow-up rate of solutions for a supercritical semilinear diffusion equation,, J. Differ. Equations, 205 (2004), 365.
doi: 10.1016/j.jde.2004.03.009. |
[7] |
M. Fila, M. Winkler and E. Yanagida, Convergence rate for a parabolic equation with supercritical nonlinearity,, J. Dynam. Differ. Equations, 17 (2005), 249.
doi: 10.1007/s10884-005-5405-2. |
[8] |
M. Fila, M. Winkler and E. Yanagida, Slow convergence to zero for a parabolic equation with supercritical nonlinearity,, Math. Annalen, 340 (2008), 477.
doi: 10.1007/s00208-007-0148-5. |
[9] |
H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha} $,, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109.
|
[10] |
V. A. Galaktionov and J. R. King, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents,, J. Differ. Equations, 189 (2003), 199.
doi: 10.1016/S0022-0396(02)00151-1. |
[11] |
V. Galaktionov and J. L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions,, Comm. Pure Applied Math., 50 (1997), 1.
doi: 10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.3.CO;2-R. |
[12] |
C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbbR^n$,, Comm. Pure Appl. Math., 45 (1992), 1153.
doi: 10.1002/cpa.3160450906. |
[13] |
C. Gui, W.-M. Ni and X. Wang, Further study on a nonlinear heat equation,, J. Differ. Equations, 169 (2001), 588.
doi: 10.1006/jdeq.2000.3909. |
[14] |
M. Hoshino and E. Yanagida, Sharp estimates of the convergence rate for a semilinear parabolic equation with supercritical nonlinearity,, Nonlin. Anal. TMA, 69 (2008), 3136.
doi: 10.1016/j.na.2007.09.007. |
[15] |
R. Ikehata, M. Ishiwata and T. Suzuki, Semilinear parabolic equation in $R^N$ associated with critical Sobolev exponent,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 69 (2010), 877.
doi: 10.1016/j.anihpc.2010.01.002. |
[16] |
M. Ishiwata, On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent,, J. Differ. Equations, 249 (2010), 1466.
doi: 10.1016/j.jde.2010.06.024. |
[17] |
O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423.
|
[18] |
T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 1.
|
[19] |
H. Matano and F. Merle, Classification of type I and type II behaviors for a supercritical nonlinear heat equation,, J. Funct. Anal., 256 (2009), 992.
doi: 10.1016/j.jfa.2008.05.021. |
[20] |
H. Matano and F. Merle, Threshold and generic type I behaviors for a supercritical nonlinear heat equation,, J. Funct. Anal., 261 (2011), 716.
doi: 10.1016/j.jfa.2011.02.025. |
[21] |
N. Mizoguchi, On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity,, Math. Z., 239 (2002), 215.
doi: 10.1007/s002090100292. |
[22] |
N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its applications,, Indiana Univ. Math. J., 54 (2005), 1047.
doi: 10.1512/iumj.2005.54.2694. |
[23] |
W.-M. Ni, P. E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations,, J. Differ. Equations, 54 (1984), 97.
doi: 10.1016/0022-0396(84)90145-1. |
[24] |
P. Poláčik and P. Quittner, Asymptotic behavior of threshold and sub-threshold solutions of a semilinear heat equation,, Asymptotic Analysis, 57 (2008), 125.
|
[25] |
P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation,, Math. Annalen, 327 (2003), 745.
doi: 10.1007/s00208-003-0469-y. |
[26] |
P. Poláčik and E. Yanagida, Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation,, Diff. Int. Equations, 17 (2004), 535.
|
[27] |
P. Quittner, The decay of global solutions of a semilinear heat equation,, Discrete Contin. Dynam. Systems A, 21 (2008), 307.
doi: 10.3934/dcds.2008.21.307. |
[28] |
P. Quittner and P. Souplet, "Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2007).
|
[29] |
Ph. Souplet, Sur l'asymptotique des solutions globales pour une équation de la chaleur semi-linéaire dans des domaines non bornés,, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 877.
|
[30] |
C. Stinner, Very slow convergence to zero for a supercritical semilinear parabolic equation,, Adv. Differ. Equations, 14 (2009), 1085.
|
[31] |
C. Stinner, Very slow convergence rates in a semilinear parabolic equation,, NoDEA, 17 (2010), 213.
doi: 10.1007/s00030-009-0050-9. |
[32] |
C. Stinner, The convergence rate for a semilinear parabolic equation with a critical exponent,, Appl. Math. Letters, 24 (2011), 454.
doi: 10.1016/j.aml.2010.10.041. |
[33] |
X. Wang, On the Cauchy problem for reaction-diffusion equations,, Trans. Amer. Math. Soc., 337 (1993), 549.
doi: 10.2307/2154232. |
show all references
References:
[1] |
M. Fila, J. R. King, M. Winkler and E. Yanagida, Optimal lower bound of the grow-up rate for a supercritical parabolic equation,, J. Differ. Equations, 228 (2006), 339.
doi: 10.1016/j.jde.2006.01.019. |
[2] |
M. Fila, J. R. King, M. Winkler and E. Yanagida, Grow-up rate of solutions of a semilinear parabolic equation with a critical exponent,, Adv. Differ. Equations, 12 (2007), 1.
|
[3] |
M. Fila, J. R. King, M. Winkler and E. Yanagida, Very slow grow-up of solutions of a semi-linear parabolic equation,, Proc. Edinb. Math. Soc., 53 (2011), 1.
doi: 10.1017/S0013091509001497. |
[4] |
M. Fila, H. Matano and P. Poláčik, Immediate regularization after blow-up,, SIAM J. Math. Anal., 37 (2005), 752.
doi: 10.1137/040613299. |
[5] |
M. Fila and M. Winkler, Rate of convergence to a singular steady state of a supercritical parabolic equation,, J. Evol. Equations, 8 (2008), 673.
doi: 10.1007/s00028-008-0400-9. |
[6] |
M. Fila, M. Winkler and E. Yanagida, Grow-up rate of solutions for a supercritical semilinear diffusion equation,, J. Differ. Equations, 205 (2004), 365.
doi: 10.1016/j.jde.2004.03.009. |
[7] |
M. Fila, M. Winkler and E. Yanagida, Convergence rate for a parabolic equation with supercritical nonlinearity,, J. Dynam. Differ. Equations, 17 (2005), 249.
doi: 10.1007/s10884-005-5405-2. |
[8] |
M. Fila, M. Winkler and E. Yanagida, Slow convergence to zero for a parabolic equation with supercritical nonlinearity,, Math. Annalen, 340 (2008), 477.
doi: 10.1007/s00208-007-0148-5. |
[9] |
H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_t=\Delta u+u^{1+\alpha} $,, J. Fac. Sci. Univ. Tokyo Sect. I, 13 (1966), 109.
|
[10] |
V. A. Galaktionov and J. R. King, Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents,, J. Differ. Equations, 189 (2003), 199.
doi: 10.1016/S0022-0396(02)00151-1. |
[11] |
V. Galaktionov and J. L. Vázquez, Continuation of blow-up solutions of nonlinear heat equations in several space dimensions,, Comm. Pure Applied Math., 50 (1997), 1.
doi: 10.1002/(SICI)1097-0312(199701)50:1<1::AID-CPA1>3.3.CO;2-R. |
[12] |
C. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat equation in $\mathbbR^n$,, Comm. Pure Appl. Math., 45 (1992), 1153.
doi: 10.1002/cpa.3160450906. |
[13] |
C. Gui, W.-M. Ni and X. Wang, Further study on a nonlinear heat equation,, J. Differ. Equations, 169 (2001), 588.
doi: 10.1006/jdeq.2000.3909. |
[14] |
M. Hoshino and E. Yanagida, Sharp estimates of the convergence rate for a semilinear parabolic equation with supercritical nonlinearity,, Nonlin. Anal. TMA, 69 (2008), 3136.
doi: 10.1016/j.na.2007.09.007. |
[15] |
R. Ikehata, M. Ishiwata and T. Suzuki, Semilinear parabolic equation in $R^N$ associated with critical Sobolev exponent,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 69 (2010), 877.
doi: 10.1016/j.anihpc.2010.01.002. |
[16] |
M. Ishiwata, On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent,, J. Differ. Equations, 249 (2010), 1466.
doi: 10.1016/j.jde.2010.06.024. |
[17] |
O. Kavian, Remarks on the large time behaviour of a nonlinear diffusion equation,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 4 (1987), 423.
|
[18] |
T. Kawanago, Asymptotic behavior of solutions of a semilinear heat equation with subcritical nonlinearity,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 13 (1996), 1.
|
[19] |
H. Matano and F. Merle, Classification of type I and type II behaviors for a supercritical nonlinear heat equation,, J. Funct. Anal., 256 (2009), 992.
doi: 10.1016/j.jfa.2008.05.021. |
[20] |
H. Matano and F. Merle, Threshold and generic type I behaviors for a supercritical nonlinear heat equation,, J. Funct. Anal., 261 (2011), 716.
doi: 10.1016/j.jfa.2011.02.025. |
[21] |
N. Mizoguchi, On the behavior of solutions for a semilinear parabolic equation with supercritical nonlinearity,, Math. Z., 239 (2002), 215.
doi: 10.1007/s002090100292. |
[22] |
N. Mizoguchi, Boundedness of global solutions for a supercritical semilinear heat equation and its applications,, Indiana Univ. Math. J., 54 (2005), 1047.
doi: 10.1512/iumj.2005.54.2694. |
[23] |
W.-M. Ni, P. E. Sacks and J. Tavantzis, On the asymptotic behavior of solutions of certain quasilinear parabolic equations,, J. Differ. Equations, 54 (1984), 97.
doi: 10.1016/0022-0396(84)90145-1. |
[24] |
P. Poláčik and P. Quittner, Asymptotic behavior of threshold and sub-threshold solutions of a semilinear heat equation,, Asymptotic Analysis, 57 (2008), 125.
|
[25] |
P. Poláčik and E. Yanagida, On bounded and unbounded global solutions of a supercritical semilinear heat equation,, Math. Annalen, 327 (2003), 745.
doi: 10.1007/s00208-003-0469-y. |
[26] |
P. Poláčik and E. Yanagida, Nonstabilizing solutions and grow-up set for a supercritical semilinear diffusion equation,, Diff. Int. Equations, 17 (2004), 535.
|
[27] |
P. Quittner, The decay of global solutions of a semilinear heat equation,, Discrete Contin. Dynam. Systems A, 21 (2008), 307.
doi: 10.3934/dcds.2008.21.307. |
[28] |
P. Quittner and P. Souplet, "Superlinear Parabolic Problems, Blow-up, Global Existence and Steady States,", Birkhäuser Advanced Texts: Basler Lehrbücher, (2007).
|
[29] |
Ph. Souplet, Sur l'asymptotique des solutions globales pour une équation de la chaleur semi-linéaire dans des domaines non bornés,, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 877.
|
[30] |
C. Stinner, Very slow convergence to zero for a supercritical semilinear parabolic equation,, Adv. Differ. Equations, 14 (2009), 1085.
|
[31] |
C. Stinner, Very slow convergence rates in a semilinear parabolic equation,, NoDEA, 17 (2010), 213.
doi: 10.1007/s00030-009-0050-9. |
[32] |
C. Stinner, The convergence rate for a semilinear parabolic equation with a critical exponent,, Appl. Math. Letters, 24 (2011), 454.
doi: 10.1016/j.aml.2010.10.041. |
[33] |
X. Wang, On the Cauchy problem for reaction-diffusion equations,, Trans. Amer. Math. Soc., 337 (1993), 549.
doi: 10.2307/2154232. |
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