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

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

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.
Citation: Juraj Földes, Peter Poláčik. On asymptotically symmetric parabolic equations. Networks and 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.

[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.

[1]

Yan Deng, Junfang Zhao, Baozeng Chu. Symmetry of positive solutions for systems of fractional Hartree equations. Discrete and 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 and 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 and 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 and Continuous Dynamical Systems, 2009, 25 (1) : 133-157. doi: 10.3934/dcds.2009.25.133

[5]

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 and Imaging, 2016, 10 (4) : 869-898. doi: 10.3934/ipi.2016025

[6]

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

[7]

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

[8]

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

[9]

Chunpeng Wang. Boundary behavior and asymptotic behavior of solutions to a class of parabolic equations with boundary degeneracy. Discrete and 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 and 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 and 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 and Applied Analysis, 2013, 12 (6) : 2685-2696. doi: 10.3934/cpaa.2013.12.2685

[13]

Xueying Chen, Guanfeng Li, Sijia Bao. Symmetry and monotonicity of positive solutions for a class of general pseudo-relativistic systems. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1755-1772. doi: 10.3934/cpaa.2022045

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

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 and Continuous Dynamical Systems, 2010, 26 (2) : 737-765. doi: 10.3934/dcds.2010.26.737

[20]

Xinxin Jing, Yuanyuan Nie, Chunpeng Wang. Asymptotic behavior of solutions to coupled semilinear parabolic equations with general degenerate diffusion coefficients. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022107

2021 Impact Factor: 1.41

Metrics

  • PDF downloads (89)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]