# American Institute of Mathematical Sciences

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 & 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.   Google Scholar [2] A. V. Babin, Symmetrization properties of parabolic equations in symmetric domains,, J. Dynam. Differential Equations, 6 (1994), 639.  doi: 10.1007/BF02218852.  Google Scholar [3] _______, Symmetry of instabilities for scalar equations in symmetric domains,, J. Differential Equations, 123 (1995), 122.  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.  doi: 10.1006/jdeq.1999.3654.  Google Scholar [5] H. Berestycki, Qualitative properties of positive solutions of elliptic equations,, Partial differential equations (Praha, 406 (2000), 34.   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.  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.  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.  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.  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.  doi: 10.1017/S0004972700012089.  Google Scholar [11] J. Földes, On symmetry properties of parabolic equations in bounded domains,, J. Differential Equations, 250 (2011), 4236.  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.  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.   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.  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.  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, 406 (2000), 214.   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.  doi: 10.1080/03605309108820766.  Google Scholar [18] W.-M. Ni, Qualitative properties of solutions to elliptic problems,, Stationary partial differential equations. Vol. I, (2004), 157.  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.  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, (2009), 170.  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.  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), 29 (2012), 1.   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.   Google Scholar

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##### References:
 [1] A. D. Alexandrov, A characteristic property of spheres,, Ann. Mat. Pura Appl. (4), 58 (1962), 303.   Google Scholar [2] A. V. Babin, Symmetrization properties of parabolic equations in symmetric domains,, J. Dynam. Differential Equations, 6 (1994), 639.  doi: 10.1007/BF02218852.  Google Scholar [3] _______, Symmetry of instabilities for scalar equations in symmetric domains,, J. Differential Equations, 123 (1995), 122.  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.  doi: 10.1006/jdeq.1999.3654.  Google Scholar [5] H. Berestycki, Qualitative properties of positive solutions of elliptic equations,, Partial differential equations (Praha, 406 (2000), 34.   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.  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.  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.  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.  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.  doi: 10.1017/S0004972700012089.  Google Scholar [11] J. Földes, On symmetry properties of parabolic equations in bounded domains,, J. Differential Equations, 250 (2011), 4236.  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.  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.   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.  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.  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, 406 (2000), 214.   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.  doi: 10.1080/03605309108820766.  Google Scholar [18] W.-M. Ni, Qualitative properties of solutions to elliptic problems,, Stationary partial differential equations. Vol. I, (2004), 157.  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.  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, (2009), 170.  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.  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), 29 (2012), 1.   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.   Google Scholar
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