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, 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), 303-315. doi: 10.1007/BF02413056.  Google Scholar [2] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar [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), 83-97. doi: 10.1016/j.anihpc.2004.05.002.  Google Scholar [4] K. Bogdan, T. Kulczycki and M. Kwaśnicki, Estimates and structure of $\alpha$-harmonic functions, Probability Theory and Related Fields, 140 (2008), 345-381. doi: 10.1007/s00440-007-0067-0.  Google Scholar [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), 39-71. doi: 10.1017/S0308210511000175.  Google Scholar [6] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9), 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005.  Google Scholar [7] L. Caffarelli, C. H. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators, J. Amer. Math. Soc., 24 (2011), 849-869. doi: 10.1090/S0894-0347-2011-00698-X.  Google Scholar [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.  Google Scholar [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar [10] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.  Google Scholar [11] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.  Google Scholar [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), 1353-1384. doi: 10.1080/03605302.2011.562954.  Google Scholar [13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [14] S.-Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.  Google Scholar [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.  Google Scholar [16] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar [17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar [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), 1237-1262. doi: 10.1017/S0308210511000746.  Google Scholar [19] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Communications in Partial Differential Equations, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211.  Google Scholar [20] B. Gidas, W. N. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar [21] N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. I, II, III, Imperial College Press, London, 2005. doi: 10.1142/9781860947155.  Google Scholar [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.  Google Scholar [23] M. Kassmann, A new formulation of Harnack's inequality for nonlocal operators, C. R. Math. Acad. Sci. Paris, 349 (2011), 637-640. doi: 10.1016/j.crma.2011.04.014.  Google Scholar [24] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar [25] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  Google Scholar [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), 1567-1593. doi: 10.1080/03605300500299919.  Google Scholar [27] P. Poláčik, Estimates of solutions and asymptotic symmetry for parabolic equations on bounded domains, Arch. Ration. Mech. Anal., 183 (2007), 59-91. doi: 10.1007/s00205-006-0004-x.  Google Scholar [28] P. Poláčik, Symmetry properties of positive solutions of parabolic equations: A survey, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific, Hackensack, NJ, 2009, 170-208. doi: 10.1142/9789812834744_0009.  Google Scholar [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., ().   Google Scholar [30] A. de Pablo, F. Quirós, A. Rodrĺguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017.  Google Scholar [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.  Google Scholar [32] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  Google Scholar [33] M. E. Schonbek and T. P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal., 35 (2003), 357-375. doi: 10.1137/S0036141002409362.  Google Scholar [34] R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory and Related Fields, 125 (2003), 578-592. doi: 10.1007/s00440-002-0251-1.  Google Scholar [35] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.  Google Scholar [36] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922.  Google Scholar

show all references

References:
 [1] A. D. Alexandrov, A characteristic property of the spheres, Ann. Mat. Pura Appl. (4), 58 (1962), 303-315. doi: 10.1007/BF02413056.  Google Scholar [2] D. Applebaum, Lévy Processes and Stochastic Calculus, 2nd edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. doi: 10.1017/CBO9780511809781.  Google Scholar [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), 83-97. doi: 10.1016/j.anihpc.2004.05.002.  Google Scholar [4] K. Bogdan, T. Kulczycki and M. Kwaśnicki, Estimates and structure of $\alpha$-harmonic functions, Probability Theory and Related Fields, 140 (2008), 345-381. doi: 10.1007/s00440-007-0067-0.  Google Scholar [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), 39-71. doi: 10.1017/S0308210511000175.  Google Scholar [6] E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, J. Math. Pures Appl. (9), 86 (2006), 271-291. doi: 10.1016/j.matpur.2006.04.005.  Google Scholar [7] L. Caffarelli, C. H. Chan and A. Vasseur, Regularity theory for parabolic nonlinear integral operators, J. Amer. Math. Soc., 24 (2011), 849-869. doi: 10.1090/S0894-0347-2011-00698-X.  Google Scholar [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.  Google Scholar [9] L. Caffarelli and L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260. doi: 10.1080/03605300600987306.  Google Scholar [10] X. Cabré and J. Tan, Positive solutions of nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), 2052-2093. doi: 10.1016/j.aim.2010.01.025.  Google Scholar [11] L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math. (2), 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.  Google Scholar [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), 1353-1384. doi: 10.1080/03605302.2011.562954.  Google Scholar [13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Oxford Lecture Series in Mathematics and its Applications, 13, The Clarendon Press, Oxford University Press, New York, 1998.  Google Scholar [14] S.-Y. A. Chang and M. del Mar González, Fractional Laplacian in conformal geometry, Adv. Math., 226 (2011), 1410-1432. doi: 10.1016/j.aim.2010.07.016.  Google Scholar [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.  Google Scholar [16] W. Chen, C. Li and B. Ou, Classification of solutions for an integral equation, Comm. Pure Appl. Math., 59 (2006), 330-343. doi: 10.1002/cpa.20116.  Google Scholar [17] E. Di Nezza, G. Palatucci and E. Valdinoci, Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.  Google Scholar [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), 1237-1262. doi: 10.1017/S0308210511000746.  Google Scholar [19] M. Felsinger and M. Kassmann, Local regularity for parabolic nonlocal operators, Communications in Partial Differential Equations, 38 (2013), 1539-1573. doi: 10.1080/03605302.2013.808211.  Google Scholar [20] B. Gidas, W. N. Ni and L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys., 68 (1979), 209-243. doi: 10.1007/BF01221125.  Google Scholar [21] N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. I, II, III, Imperial College Press, London, 2005. doi: 10.1142/9781860947155.  Google Scholar [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.  Google Scholar [23] M. Kassmann, A new formulation of Harnack's inequality for nonlocal operators, C. R. Math. Acad. Sci. Paris, 349 (2011), 637-640. doi: 10.1016/j.crma.2011.04.014.  Google Scholar [24] G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.  Google Scholar [25] R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.  Google Scholar [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), 1567-1593. doi: 10.1080/03605300500299919.  Google Scholar [27] P. Poláčik, Estimates of solutions and asymptotic symmetry for parabolic equations on bounded domains, Arch. Ration. Mech. Anal., 183 (2007), 59-91. doi: 10.1007/s00205-006-0004-x.  Google Scholar [28] P. Poláčik, Symmetry properties of positive solutions of parabolic equations: A survey, in Recent Progress on Reaction-Diffusion Systems and Viscosity Solutions, World Scientific, Hackensack, NJ, 2009, 170-208. doi: 10.1142/9789812834744_0009.  Google Scholar [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., ().   Google Scholar [30] A. de Pablo, F. Quirós, A. Rodrĺguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017.  Google Scholar [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.  Google Scholar [32] J. Serrin, A symmetry problem in potential theory, Arch. Rational Mech. Anal., 43 (1971), 304-318.  Google Scholar [33] M. E. Schonbek and T. P. Schonbek, Asymptotic behavior to dissipative quasi-geostrophic flows, SIAM J. Math. Anal., 35 (2003), 357-375. doi: 10.1137/S0036141002409362.  Google Scholar [34] R. Song and Z. Vondraček, Potential theory of subordinate killed Brownian motion in a domain, Probab. Theory and Related Fields, 125 (2003), 578-592. doi: 10.1007/s00440-002-0251-1.  Google Scholar [35] J. Tan, The Brezis-Nirenberg type problem involving the square root of the Laplacian, Calc. Var. Partial Differential Equations, 42 (2011), 21-41. doi: 10.1007/s00526-010-0378-3.  Google Scholar [36] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922.  Google Scholar
 [1] 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 [2] Pablo Raúl Stinga, Chao Zhang. Harnack's inequality for fractional nonlocal equations. Discrete & Continuous Dynamical Systems, 2013, 33 (7) : 3153-3170. doi: 10.3934/dcds.2013.33.3153 [3] Jinggang Tan, Jingang Xiong. A Harnack inequality for fractional Laplace equations with lower order terms. Discrete & Continuous Dynamical Systems, 2011, 31 (3) : 975-983. doi: 10.3934/dcds.2011.31.975 [4] Fabio Paronetto. A Harnack type inequality and a maximum principle for an elliptic-parabolic and forward-backward parabolic De Giorgi class. Discrete & Continuous Dynamical Systems - S, 2017, 10 (4) : 853-866. doi: 10.3934/dcdss.2017043 [5] 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 [6] Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154 [7] Zhigang Wu, Hao Xu. Symmetry properties in systems of fractional Laplacian equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1559-1571. doi: 10.3934/dcds.2019068 [8] Brian D. Ewald, Roger Témam. Maximum principles for the primitive equations of the atmosphere. Discrete & Continuous Dynamical Systems, 2001, 7 (2) : 343-362. doi: 10.3934/dcds.2001.7.343 [9] Meixia Dou. A direct method of moving planes for fractional Laplacian equations in the unit ball. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1797-1807. doi: 10.3934/cpaa.2016015 [10] Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201 [11] Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125 [12] CÉSAR E. TORRES LEDESMA. Existence and symmetry result for fractional p-Laplacian in $\mathbb{R}^{n}$. Communications on Pure & Applied Analysis, 2017, 16 (1) : 99-114. doi: 10.3934/cpaa.2017004 [13] 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 [14] Ran Zhuo, Yan Li. Nonexistence and symmetry of solutions for Schrödinger systems involving fractional Laplacian. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1595-1611. doi: 10.3934/dcds.2019071 [15] Daniela De Silva, Ovidiu Savin. A note on higher regularity boundary Harnack inequality. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 6155-6163. doi: 10.3934/dcds.2015.35.6155 [16] Giuseppe Di Fazio, Maria Stella Fanciullo, Pietro Zamboni. Harnack inequality for degenerate elliptic equations and sum operators. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2363-2376. doi: 10.3934/cpaa.2015.14.2363 [17] Xi Wang, Zuhan Liu, Ling Zhou. Asymptotic decay for the classical solution of the chemotaxis system with fractional Laplacian in high dimensions. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 4003-4020. doi: 10.3934/dcdsb.2018121 [18] Pu-Zhao Kow, Masato Kimura. The Lewy-Stampacchia inequality for the fractional Laplacian and its application to anomalous unidirectional diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021167 [19] Phuong Le. Symmetry of singular solutions for a weighted Choquard equation involving the fractional $p$-Laplacian. Communications on Pure & Applied Analysis, 2020, 19 (1) : 527-539. doi: 10.3934/cpaa.2020026 [20] Lihong Zhang, Wenwen Hou, Bashir Ahmad, Guotao Wang. Radial symmetry for logarithmic Choquard equation involving a generalized tempered fractional $p$-Laplacian. Discrete & Continuous Dynamical Systems - S, 2021, 14 (10) : 3851-3863. doi: 10.3934/dcdss.2020445

2020 Impact Factor: 1.392