# American Institute of Mathematical Sciences

June  2014, 34(6): 2581-2615. doi: 10.3934/dcds.2014.34.2581

## Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations

 1 Institut für Mathematik, Goethe-Universität, Frankfurt, Robert-Mayer-Straße 10, 60054 Frankfurt, Germany, Germany

Received  February 2013 Published  December 2013

We study the nonlinear fractional reaction-diffusion equation $∂_t u + (-\Delta)^s u = f(t,x,u)$, $s\in(0,1)$ in a bounded domain $\Omega$ together with Dirichlet boundary conditions on $\mathbb{R}^N \setminus \Omega$. We prove asymptotic symmetry of nonnegative globally bounded solutions in the case where the underlying data obeys some symmetry and monotonicity assumptions. More precisely, we assume that $\Omega$ is symmetric with respect to reflection at a hyperplane, say $\{x_1=0\}$, and convex in the $x_1$-direction, and that the nonlinearity $f$ is even in $x_1$ and nonincreasing in $|x_1|$. Under rather weak additional technical assumptions, we then show that any nonzero element in the $\omega$-limit set of nonnegative globally bounded solution is even in $x_1$ and strictly decreasing in $|x_1|$. This result, which is obtained via a series of new estimates for antisymmetric supersolutions of a corresponding family of linear equations, implies a strong maximum type principle which is not available in the non-fractional case $s=1$.
Citation: Sven Jarohs, Tobias Weth. Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2581-2615. doi: 10.3934/dcds.2014.34.2581
##### References:
 [1] A. D. Alexandrov, A characteristic property of the spheres,, Ann. Mat. Pura Appl. (4), 58 (1962), 303. doi: 10.1007/BF02413056. [2] D. Applebaum, Lévy Processes and Stochastic Calculus,, 2nd edition, (2009). doi: 10.1017/CBO9780511809781. [3] M. Birkner, J. A. López-Mimbela and A. Wakolbinger, Comparison results and steady states for the Fujita equation with fractional Laplacian,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 83. doi: 10.1016/j.anihpc.2004.05.002. [4] K. Bogdan, T. Kulczycki and M. Kwaśnicki, Estimates and structure of $\alpha$-harmonic functions,, Probability Theory and Related Fields, 140 (2008), 345. doi: 10.1007/s00440-007-0067-0. [5] C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39. doi: 10.1017/S0308210511000175. [6] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl. (9), 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005. [7] L. Caffarelli, C. H. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators,, J. Amer. Math. Soc., 24 (2011), 849. doi: 10.1090/S0894-0347-2011-00698-X. [8] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians. I. Regularity, maximum principles, and Hamiltonian estimates,, Ann. Inst. H. Poincaré Anal. Non Linéaire, (). doi: 10.1016/j.anihpc.2013.02.001. [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. [10] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. [11] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. of Math. (2), 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903. [12] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local seminlinear equations,, Comm. Partial Differential Equations, 36 (2011), 1353. doi: 10.1080/03605302.2011.562954. [13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, (1998). [14] S.-Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry,, Adv. Math., 226 (2011), 1410. doi: 10.1016/j.aim.2010.07.016. [15] H. Chang Lara and G. Dávila, Regularity for solutions of non local parabolic equations,, Calculus of Variations and Partial Differential Equations, (2012). doi: 10.1007/s00526-012-0576-2. [16] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. [17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. [18] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237. doi: 10.1017/S0308210511000746. [19] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators,, Communications in Partial Differential Equations, 38 (2013), 1539. doi: 10.1080/03605302.2013.808211. [20] B. Gidas, W. N. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125. [21] N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. I, II, III,, Imperial College Press, (2005). doi: 10.1142/9781860947155. [22] T. Jin and J. Xiong, A fractional Yamabe flow and some applications,, Journal für die reine und angewandte Mathematik, (). doi: 10.1515/crelle-2012-0110. [23] M. Kassmann, A new formulation of Harnack's inequality for nonlocal operators,, C. R. Math. Acad. Sci. Paris, 349 (2011), 637. doi: 10.1016/j.crma.2011.04.014. [24] G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996). [25] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, Discrete Contin. Dyn. Syst., 33 (2013), 2105. [26] P. Poláčik, Symmetry properties of positive solutions of parabolic equations on $\mathbbR^N$. I. Asymptotic symmetry for the Cauchy problem,, Comm. Partial Differential Equations, 30 (2005), 1567. doi: 10.1080/03605300500299919. [27] 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. [28] P. Poláčik, Symmetry properties of positive solutions of parabolic equations: A survey,, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, (2009), 170. doi: 10.1142/9789812834744_0009. [29] P. Poláčik and S. Terracini, Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains,, to appear in Proceedings of the American Mathematical Society., (). [30] A. de Pablo, F. Quirós, A. Rodrĺguez and J. L. Vázquez, A fractional porous medium equation,, Adv. Math., 226 (2011), 1378. doi: 10.1016/j.aim.2010.07.017. [31] X. Ros-Oton and J. Serra, The Dirichlet Problem for the fractional Laplacian: Regularity up to the boundary,, Journal de Mathématiques Pures et Appliquées, (). doi: 10.1016/j.matpur.2013.06.003. [32] J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. [33] M. E. Schonbek and T. P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows,, SIAM J. Math. Anal., 35 (2003), 357. doi: 10.1137/S0036141002409362. [34] R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain,, Probab. Theory and Related Fields, 125 (2003), 578. doi: 10.1007/s00440-002-0251-1. [35] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21. doi: 10.1007/s00526-010-0378-3. [36] G. N. Watson, A Treatise on the Theory of Bessel Functions,, Cambridge University Press, (1922).

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##### References:
 [1] A. D. Alexandrov, A characteristic property of the spheres,, Ann. Mat. Pura Appl. (4), 58 (1962), 303. doi: 10.1007/BF02413056. [2] D. Applebaum, Lévy Processes and Stochastic Calculus,, 2nd edition, (2009). doi: 10.1017/CBO9780511809781. [3] M. Birkner, J. A. López-Mimbela and A. Wakolbinger, Comparison results and steady states for the Fujita equation with fractional Laplacian,, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 83. doi: 10.1016/j.anihpc.2004.05.002. [4] K. Bogdan, T. Kulczycki and M. Kwaśnicki, Estimates and structure of $\alpha$-harmonic functions,, Probability Theory and Related Fields, 140 (2008), 345. doi: 10.1007/s00440-007-0067-0. [5] C. Brändle, E. Colorado and A. de Pablo, A concave-convex elliptic problem involving the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 143 (2013), 39. doi: 10.1017/S0308210511000175. [6] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations,, J. Math. Pures Appl. (9), 86 (2006), 271. doi: 10.1016/j.matpur.2006.04.005. [7] L. Caffarelli, C. H. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators,, J. Amer. Math. Soc., 24 (2011), 849. doi: 10.1090/S0894-0347-2011-00698-X. [8] X. Cabré and Y. Sire, Nonlinear equations for fractional Laplacians. I. Regularity, maximum principles, and Hamiltonian estimates,, Ann. Inst. H. Poincaré Anal. Non Linéaire, (). doi: 10.1016/j.anihpc.2013.02.001. [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian,, Comm. Partial Differential Equations, 32 (2007), 1245. doi: 10.1080/03605300600987306. [10] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian,, Adv. Math., 224 (2010), 2052. doi: 10.1016/j.aim.2010.01.025. [11] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation,, Ann. of Math. (2), 171 (2010), 1903. doi: 10.4007/annals.2010.171.1903. [12] A. Capella, J. Dávila, L. Dupaigne and Y. Sire, Regularity of radial extremal solutions for some non-local seminlinear equations,, Comm. Partial Differential Equations, 36 (2011), 1353. doi: 10.1080/03605302.2011.562954. [13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations,, Oxford Lecture Series in Mathematics and its Applications, (1998). [14] S.-Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry,, Adv. Math., 226 (2011), 1410. doi: 10.1016/j.aim.2010.07.016. [15] H. Chang Lara and G. Dávila, Regularity for solutions of non local parabolic equations,, Calculus of Variations and Partial Differential Equations, (2012). doi: 10.1007/s00526-012-0576-2. [16] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation,, Comm. Pure Appl. Math., 59 (2006), 330. doi: 10.1002/cpa.20116. [17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces,, Bull. Sci. Math., 136 (2012), 521. doi: 10.1016/j.bulsci.2011.12.004. [18] P. Felmer, A. Quaas and J. Tan, Positive solutions of nonlinear Schrödinger equation with the fractional Laplacian,, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 1237. doi: 10.1017/S0308210511000746. [19] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators,, Communications in Partial Differential Equations, 38 (2013), 1539. doi: 10.1080/03605302.2013.808211. [20] B. Gidas, W. N. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle,, Comm. Math. Phys., 68 (1979), 209. doi: 10.1007/BF01221125. [21] N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. I, II, III,, Imperial College Press, (2005). doi: 10.1142/9781860947155. [22] T. Jin and J. Xiong, A fractional Yamabe flow and some applications,, Journal für die reine und angewandte Mathematik, (). doi: 10.1515/crelle-2012-0110. [23] M. Kassmann, A new formulation of Harnack's inequality for nonlocal operators,, C. R. Math. Acad. Sci. Paris, 349 (2011), 637. doi: 10.1016/j.crma.2011.04.014. [24] G. M. Lieberman, Second Order Parabolic Differential Equations,, World Scientific Publishing Co., (1996). [25] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type,, Discrete Contin. Dyn. Syst., 33 (2013), 2105. [26] P. Poláčik, Symmetry properties of positive solutions of parabolic equations on $\mathbbR^N$. I. Asymptotic symmetry for the Cauchy problem,, Comm. Partial Differential Equations, 30 (2005), 1567. doi: 10.1080/03605300500299919. [27] 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. [28] P. Poláčik, Symmetry properties of positive solutions of parabolic equations: A survey,, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, (2009), 170. doi: 10.1142/9789812834744_0009. [29] P. Poláčik and S. Terracini, Nonnegative solutions with a nontrivial nodal set for elliptic equations on smooth symmetric domains,, to appear in Proceedings of the American Mathematical Society., (). [30] A. de Pablo, F. Quirós, A. Rodrĺguez and J. L. Vázquez, A fractional porous medium equation,, Adv. Math., 226 (2011), 1378. doi: 10.1016/j.aim.2010.07.017. [31] X. Ros-Oton and J. Serra, The Dirichlet Problem for the fractional Laplacian: Regularity up to the boundary,, Journal de Mathématiques Pures et Appliquées, (). doi: 10.1016/j.matpur.2013.06.003. [32] J. Serrin, A symmetry problem in potential theory,, Arch. Rational Mech. Anal., 43 (1971), 304. [33] M. E. Schonbek and T. P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows,, SIAM J. Math. Anal., 35 (2003), 357. doi: 10.1137/S0036141002409362. [34] R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain,, Probab. Theory and Related Fields, 125 (2003), 578. doi: 10.1007/s00440-002-0251-1. [35] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian,, Calc. Var. Partial Differential Equations, 42 (2011), 21. doi: 10.1007/s00526-010-0378-3. [36] G. N. Watson, A Treatise on the Theory of Bessel Functions,, Cambridge University Press, (1922).
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