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On asymptotically symmetric parabolic equations
1. | Institute for Mathematics and its Applicaitons, University of Minnesota, Minneapolis, MN 55455, United States |
2. | School of Mathematics, University of Minnesota, Minneapolis, MN 55455 |
References:
[1] |
A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4), 58 (1962), 303-315. |
[2] |
A. V. Babin, Symmetrization properties of parabolic equations in symmetric domains, J. Dynam. Differential Equations, 6 (1994), 639-658.
doi: 10.1007/BF02218852. |
[3] |
_______, Symmetry of instabilities for scalar equations in symmetric domains, J. Differential Equations, 123 (1995), 122-152.
doi: 10.1006/jdeq.1995.1159. |
[4] |
A. V. Babin and G. R. Sell, Attractors of non-autonomous parabolic equations and their symmetry properties, J. Differential Equations, 160 (2000), 1-50.
doi: 10.1006/jdeq.1999.3654. |
[5] |
H. Berestycki, Qualitative properties of positive solutions of elliptic equations, Partial differential equations (Praha, 1998), Chapman & Hall/CRC Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, FL, (2000), 34-44. |
[6] |
H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[7] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[8] |
X. Cabré, On the Alexandroff-Bakel'man-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48 (1995), 539-570.
doi: 10.1002/cpa.3160480504. |
[9] |
F. Da Lio and B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. European Math. Soc., 9 (2007), 317-330.
doi: 10.4171/JEMS/81. |
[10] |
E. N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc., 46 (1992), 425-434.
doi: 10.1017/S0004972700012089. |
[11] |
J. Földes, On symmetry properties of parabolic equations in bounded domains, J. Differential Equations, 250 (2011), 4236-4261.
doi: 10.1016/j.jde.2011.03.018. |
[12] |
_____, Symmetry of positive solutions of asymptotically symmetric parabolic problems on $\mathbb{R}^N2$, J. Dynam. Differential Equations, 23 (2011), 45-69.
doi: 10.1007/s10884-010-9193-y. |
[13] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. |
[14] |
P. Hess and P. Poláčik, Symmetry and convergence properties for non-negative solutions of nonautonomous reaction-diffusion problems, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 573-587.
doi: 10.1017/S030821050002878X. |
[15] |
J. Jiang and J Shi, Dynamics of a reaction-diffusion system of autocatalytic chemical reaction, Discrete Contin. Dynam. Systems, 21 (2008), 245-258.
doi: 10.3934/dcds.2008.21.245. |
[16] |
B. Kawohl, Symmetrization-or how to prove symmetry of solutions to a PDE, Partial differential equations (Praha, 1998), Chapman & Hall/CRC Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, FL, (2000), 214-229. |
[17] |
C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Comm. Partial Differential Equations, 16 (1991), 491-526.
doi: 10.1080/03605309108820766. |
[18] |
W.-M. Ni, Qualitative properties of solutions to elliptic problems, Stationary partial differential equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 157-233.
doi: 10.1016/S1874-5733(04)80005-6. |
[19] |
P. Poláčik, Estimates of solutions and asymptotic symmetry for parabolic equations on bounded domains, Arch. Ration. Mech. Anal., 183 (2007), 59-91. Addendum: www.math.umn.edu/ polacik/Publications.
doi: 10.1007/s00205-006-0004-x. |
[20] |
_______, Symmetry properties of positive solutions of parabolic equations: A survey, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, (2009), 170-208.
doi: 10.1142/9789812834744_0009. |
[21] |
________, Symmetry of nonnegative solutions of elliptic equations via a result of Serrin, Comm. Partial Differential Equations, 36 (2011), 657-669.
doi: 10.1080/03605302.2010.513026. |
[22] |
________, On symmetry of nonnegative solutions of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Lineaire 29 (2012), 1-19. |
[23] |
A. Saldaña and T. Weth, Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains, J. Evol. Equ. (to appear).
doi: 10.1007/s00028-012-0150-6. |
[24] |
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. |
show all references
References:
[1] |
A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4), 58 (1962), 303-315. |
[2] |
A. V. Babin, Symmetrization properties of parabolic equations in symmetric domains, J. Dynam. Differential Equations, 6 (1994), 639-658.
doi: 10.1007/BF02218852. |
[3] |
_______, Symmetry of instabilities for scalar equations in symmetric domains, J. Differential Equations, 123 (1995), 122-152.
doi: 10.1006/jdeq.1995.1159. |
[4] |
A. V. Babin and G. R. Sell, Attractors of non-autonomous parabolic equations and their symmetry properties, J. Differential Equations, 160 (2000), 1-50.
doi: 10.1006/jdeq.1999.3654. |
[5] |
H. Berestycki, Qualitative properties of positive solutions of elliptic equations, Partial differential equations (Praha, 1998), Chapman & Hall/CRC Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, FL, (2000), 34-44. |
[6] |
H. Berestycki and L. Nirenberg, On the method of moving planes and the sliding method, Bol. Soc. Brasil. Mat. (N.S.), 22 (1991), 1-37.
doi: 10.1007/BF01244896. |
[7] |
H. Berestycki, L. Nirenberg and S. R. S. Varadhan, The principal eigenvalue and maximum principle for second-order elliptic operators in general domains, Comm. Pure Appl. Math., 47 (1994), 47-92.
doi: 10.1002/cpa.3160470105. |
[8] |
X. Cabré, On the Alexandroff-Bakel'man-Pucci estimate and the reversed Hölder inequality for solutions of elliptic and parabolic equations, Comm. Pure Appl. Math., 48 (1995), 539-570.
doi: 10.1002/cpa.3160480504. |
[9] |
F. Da Lio and B. Sirakov, Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations, J. European Math. Soc., 9 (2007), 317-330.
doi: 10.4171/JEMS/81. |
[10] |
E. N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc., 46 (1992), 425-434.
doi: 10.1017/S0004972700012089. |
[11] |
J. Földes, On symmetry properties of parabolic equations in bounded domains, J. Differential Equations, 250 (2011), 4236-4261.
doi: 10.1016/j.jde.2011.03.018. |
[12] |
_____, Symmetry of positive solutions of asymptotically symmetric parabolic problems on $\mathbb{R}^N2$, J. Dynam. Differential Equations, 23 (2011), 45-69.
doi: 10.1007/s10884-010-9193-y. |
[13] |
B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. |
[14] |
P. Hess and P. Poláčik, Symmetry and convergence properties for non-negative solutions of nonautonomous reaction-diffusion problems, Proc. Roy. Soc. Edinburgh Sect. A, 124 (1994), 573-587.
doi: 10.1017/S030821050002878X. |
[15] |
J. Jiang and J Shi, Dynamics of a reaction-diffusion system of autocatalytic chemical reaction, Discrete Contin. Dynam. Systems, 21 (2008), 245-258.
doi: 10.3934/dcds.2008.21.245. |
[16] |
B. Kawohl, Symmetrization-or how to prove symmetry of solutions to a PDE, Partial differential equations (Praha, 1998), Chapman & Hall/CRC Res. Notes Math., 406, Chapman & Hall/CRC, Boca Raton, FL, (2000), 214-229. |
[17] |
C. Li, Monotonicity and symmetry of solutions of fully nonlinear elliptic equations on bounded domains, Comm. Partial Differential Equations, 16 (1991), 491-526.
doi: 10.1080/03605309108820766. |
[18] |
W.-M. Ni, Qualitative properties of solutions to elliptic problems, Stationary partial differential equations. Vol. I, Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 157-233.
doi: 10.1016/S1874-5733(04)80005-6. |
[19] |
P. Poláčik, Estimates of solutions and asymptotic symmetry for parabolic equations on bounded domains, Arch. Ration. Mech. Anal., 183 (2007), 59-91. Addendum: www.math.umn.edu/ polacik/Publications.
doi: 10.1007/s00205-006-0004-x. |
[20] |
_______, Symmetry properties of positive solutions of parabolic equations: A survey, Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Sci. Publ., Hackensack, NJ, (2009), 170-208.
doi: 10.1142/9789812834744_0009. |
[21] |
________, Symmetry of nonnegative solutions of elliptic equations via a result of Serrin, Comm. Partial Differential Equations, 36 (2011), 657-669.
doi: 10.1080/03605302.2010.513026. |
[22] |
________, On symmetry of nonnegative solutions of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Lineaire 29 (2012), 1-19. |
[23] |
A. Saldaña and T. Weth, Asymptotic axial symmetry of solutions of parabolic equations in bounded radial domains, J. Evol. Equ. (to appear).
doi: 10.1007/s00028-012-0150-6. |
[24] |
J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318. |
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