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

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

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

Sven Jarohs ^1, and Tobias Weth ^1,

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

Received February 2013 Published December 2013

Abstract
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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 $s=1$.

Keywords: Fractional Laplacian, asymptotic symmetry, moving hyperplanes, Harnack inequality, maximum principles..

Mathematics Subject Classification: Primary: 35K58; Secondary: 35B5.

Citation: 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

References:

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[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.
[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.
[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.
[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.
[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.
[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.
[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), 1245-1260. 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), 2052-2093. 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), 1903-1930. 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), 1353-1384. 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, 13, The Clarendon Press, Oxford University Press, New York, 1998.
[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.
[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), 330-343. 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), 521-573. 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), 1237-1262. doi: 10.1017/S0308210511000746.
[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.
[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.
[21]	N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. I, II, III, Imperial College Press, London, 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), 637-640. doi: 10.1016/j.crma.2011.04.014.
[24]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
[25]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
[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.
[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.
[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.
[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), 1378-1409. 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), 304-318.
[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.
[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.
[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.
[36]	G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922.

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.
[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.
[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.
[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.
[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.
[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.
[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.
[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), 1245-1260. 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), 2052-2093. 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), 1903-1930. 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), 1353-1384. 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, 13, The Clarendon Press, Oxford University Press, New York, 1998.
[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.
[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), 330-343. 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), 521-573. 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), 1237-1262. doi: 10.1017/S0308210511000746.
[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.
[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.
[21]	N. Jacob, Pseudo Differential Operators and Markov Processes. Vol. I, II, III, Imperial College Press, London, 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), 637-640. doi: 10.1016/j.crma.2011.04.014.
[24]	G. M. Lieberman, Second Order Parabolic Differential Equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996.
[25]	R. Servadei and E. Valdinoci, Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. Syst., 33 (2013), 2105-2137.
[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.
[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.
[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.
[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), 1378-1409. 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), 304-318.
[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.
[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.
[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.
[36]	G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, 1922.

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