American Institute of Mathematical Sciences

Advanced
December  2012, 7(4): 673-689. doi: 10.3934/nhm.2012.7.673

On asymptotically symmetric parabolic equations

Juraj Földes 1, and  Peter Poláčik 2,

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

Received  January 2012 Revised  June 2012 Published  December 2012

We consider global bounded solutions of fully nonlinear parabolic equations on bounded reflectionally symmetric domains, under nonhomogeneous Dirichlet boundary condition. We assume that, as $t→∞$, the equation is asymptotically symmetric, the boundary condition is asymptotically homogeneous, and the solution is asymptotically strictly positive in the sense that all its limit profiles are strictly positive. Our main theorem states that all the limit profiles are reflectionally symmetric and decreasing on one side of the symmetry hyperplane in the direction perpendicular to the hyperplane. We also illustrate by example that, unlike for equations which are symmetric at all finite times, the result does not hold under a relaxed positivity condition requiring merely that at least one limit profile of the solution be strictly positive.
Keywords: Asymptotic symmetry, parabolic equations., positive solutions.
Mathematics Subject Classification: Primary: 35B40, 35B06; Secondary: 35K5.
Citation: Juraj Földes, Peter Poláčik. On asymptotically symmetric parabolic equations. Networks & Heterogeneous Media, 2012, 7 (4) : 673-689. doi: 10.3934/nhm.2012.7.673
References:
[1]

A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4), 58 (1962), 303-315.  Google Scholar

[2]

A. V. Babin, Symmetrization properties of parabolic equations in symmetric domains, J. Dynam. Differential Equations, 6 (1994), 639-658. doi: 10.1007/BF02218852.  Google Scholar

[3]

_______, Symmetry of instabilities for scalar equations in symmetric domains, J. Differential Equations, 123 (1995), 122-152. doi: 10.1006/jdeq.1995.1159.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc., 46 (1992), 425-434. doi: 10.1017/S0004972700012089.  Google Scholar

[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.  Google Scholar

[12]

_____, Symmetry of positive solutions of asymptotically symmetric parabolic problems on $\mathbbR^N$, J. Dynam. Differential Equations, 23 (2011), 45-69. doi: 10.1007/s10884-010-9193-y.  Google Scholar

[13]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

________, On symmetry of nonnegative solutions of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Lineaire 29 (2012), 1-19.  Google Scholar

[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.  Google Scholar

[24]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  Google Scholar

show all references

References:
[1]

A. D. Alexandrov, A characteristic property of spheres, Ann. Mat. Pura Appl. (4), 58 (1962), 303-315.  Google Scholar

[2]

A. V. Babin, Symmetrization properties of parabolic equations in symmetric domains, J. Dynam. Differential Equations, 6 (1994), 639-658. doi: 10.1007/BF02218852.  Google Scholar

[3]

_______, Symmetry of instabilities for scalar equations in symmetric domains, J. Differential Equations, 123 (1995), 122-152. doi: 10.1006/jdeq.1995.1159.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

E. N. Dancer, Some notes on the method of moving planes, Bull. Austral. Math. Soc., 46 (1992), 425-434. doi: 10.1017/S0004972700012089.  Google Scholar

[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.  Google Scholar

[12]

_____, Symmetry of positive solutions of asymptotically symmetric parabolic problems on $\mathbbR^N$, J. Dynam. Differential Equations, 23 (2011), 45-69. doi: 10.1007/s10884-010-9193-y.  Google Scholar

[13]

B. Gidas, W. M. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

________, On symmetry of nonnegative solutions of elliptic equations, Ann. Inst. H. Poincaré Anal. Non Lineaire 29 (2012), 1-19.  Google Scholar

[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.  Google Scholar

[24]

J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  Google Scholar

[1]

Yan Deng, Junfang Zhao, Baozeng Chu. Symmetry of positive solutions for systems of fractional Hartree equations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (9) : 3085-3096. doi: 10.3934/dcdss.2021079
[2]

Leyun Wu, Pengcheng Niu. Symmetry and nonexistence of positive solutions to fractional p-Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1573-1583. doi: 10.3934/dcds.2019069
[3]

Yunyun Hu. Symmetry of positive solutions to fractional equations in bounded domains and unbounded cylinders. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3723-3734. doi: 10.3934/cpaa.2020164
[4]

J. Földes, Peter Poláčik. On cooperative parabolic systems: Harnack inequalities and asymptotic symmetry. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 133-157. doi: 10.3934/dcds.2009.25.133
[5]

Shinji Adachi, Masataka Shibata, Tatsuya Watanabe. Asymptotic behavior of positive solutions for a class of quasilinear elliptic equations with general nonlinearities. Communications on Pure & Applied Analysis, 2014, 13 (1) : 97-118. doi: 10.3934/cpaa.2014.13.97
[6]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. II. Convergence of the method of finite differences. Inverse Problems & Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025
[7]

Peter Poláčik. On uniqueness of positive entire solutions and other properties of linear parabolic equations. Discrete & Continuous Dynamical Systems, 2005, 12 (1) : 13-26. doi: 10.3934/dcds.2005.12.13
[8]

Pei Ma, Yan Li, Jihui Zhang. Symmetry and nonexistence of positive solutions for fractional systems. Communications on Pure & Applied Analysis, 2018, 17 (3) : 1053-1070. doi: 10.3934/cpaa.2018051
[9]

Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1041-1060. doi: 10.3934/dcds.2016.36.1041
[10]

P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151
[11]

Ugur G. Abdulla. On the optimal control of the free boundary problems for the second order parabolic equations. I. Well-posedness and convergence of the method of lines. Inverse Problems & Imaging, 2013, 7 (2) : 307-340. doi: 10.3934/ipi.2013.7.307
[12]

Shiren Zhu, Xiaoli Chen, Jianfu Yang. Regularity, symmetry and uniqueness of positive solutions to a nonlinear elliptic system. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685
[13]

J. Húska, Peter Poláčik, M.V. Safonov. Principal eigenvalues, spectral gaps and exponential separation between positive and sign-changing solutions of parabolic equations. Conference Publications, 2005, 2005 (Special) : 427-435. doi: 10.3934/proc.2005.2005.427
[14]

Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581
[15]

Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393
[16]

Hwai-Chiuan Wang. Stability and symmetry breaking of solutions of semilinear elliptic equations. Conference Publications, 2005, 2005 (Special) : 886-894. doi: 10.3934/proc.2005.2005.886
[17]

Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
[18]

Junde Wu, Shangbin Cui. Asymptotic behavior of solutions for parabolic differential equations with invariance and applications to a free boundary problem modeling tumor growth. Discrete & Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737
[19]

Minkyu Kwak, Kyong Yu. The asymptotic behavior of solutions of a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 1996, 2 (4) : 483-496. doi: 10.3934/dcds.1996.2.483
[20]

Shota Sato, Eiji Yanagida. Asymptotic behavior of singular solutions for a semilinear parabolic equation. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 4027-4043. doi: 10.3934/dcds.2012.32.4027

2020 Impact Factor: 1.213

Tools

Metrics

  • PDF downloads (66)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences