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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 |
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. |
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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