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Dynamics of a multigroup SIRS epidemic model with random perturbations and varying total population size
Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres
Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaan xi, 710129, China |
$ (-\Delta)^{\frac{\alpha}{2}}\vec {u}(x) = \vec {h}(x,\vec {u}(x)), $ |
$ 0<\alpha <2 $ |
$ \vec u $ |
$ \vec h $ |
$ k $ |
$ \vec {h}(x,\vec {u}(x)) $ |
$ \vec {u} $ |
References:
[1] |
D. Applebeaum, Lévy Processes and Stochastic Calculus, second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. |
[2] |
J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972. |
[3] |
M. Belloni, V. Ferone and B. Kawohl,
Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Special issue dedicated to Lawrence E. Payne. Z. Angew. Math. Phys., 54 (2003), 771-783.
doi: 10.1007/s00033-003-3209-y. |
[4] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics 121, Cambridge University Press, Cambridge, 1996. |
[5] |
P. J. Bouchard and A. Georges,
Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications, Phys. Rep., 195 (1990), 127-293.
|
[6] |
C. Brandle, E. Colorado, A. de Pablo and U. Sanchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect., 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[7] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Math., 224 (2010), 2052-2053.
doi: 10.1016/j.aim.2010.01.025. |
[8] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Patial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[9] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[10] |
L. Cao and W. Chen,
Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955.
doi: 10.3934/dcds.2013.33.3937. |
[11] |
W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017).
doi: 10.1007/s00526-017-1110-3. |
[12] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Advances in Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[13] |
W. Chen, Y. Li and R. Zhang,
A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.
doi: 10.1016/j.jfa.2017.02.022. |
[14] |
W. Chen and J. Zhu,
Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2001), 744-753.
doi: 10.1016/j.jmaa.2010.11.035. |
[15] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, Lecture notes in Math. Springer, Berlin, 1871 (2006), 1–43.
doi: 10.1007/11545989_1. |
[16] |
M. Fall and T. Weth,
Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.
doi: 10.1016/j.jfa.2012.06.018. |
[17] |
S. Jarohs and T. Weth,
Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl., 195 (2016), 273-291.
doi: 10.1007/s10231-014-0462-y. |
[18] |
D. Li, P. Niu and R. Zhuo,
Nonexistence of positive solutions for an integral equation related to the Hardy-Sobolev inequality, Acta Appl. Math., 134 (2014), 185-200.
doi: 10.1007/s10440-014-9878-z. |
[19] |
D. Li, P. Niu and R. Zhuo, Symmetry and non-existence of positive solutions for PDE system with Navier boundary conditions on a half space, Compex Var. Elliptic Equ., 59 (2014).
doi: 10.1080/17476933.2013.854346. |
[20] |
D. Li, P. Niu and R. Zhuo,
Symmetry and nonexistence of positive solutions of integral systems with hardy term, J. Math. Anal. Appl., 424 (2015), 915-931.
doi: 10.1016/j.jmaa.2014.11.029. |
[21] |
G. Lu and J. Zhu,
An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.
doi: 10.1016/j.na.2011.11.036. |
[22] |
W. Reichel and H. Zou,
Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations, 161 (2000), 219-243.
doi: 10.1006/jdeq.1999.3700. |
[23] |
V. Tarasov and G. Zaslasvky,
Fractional dynamics of systems with long-range interaction, Comm. Nonl Sci. Numer. Simul., 11 (2006), 885-898.
doi: 10.1016/j.cnsns.2006.03.005. |
[24] |
P. Wang and P. Niu,
A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pur. Appl. Anal., 16 (2017), 1707-1718.
|
[25] |
P. Wang and M. Yu,
Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl., 450 (2017), 982-995.
|
[26] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
[27] |
R. Zhuo and D. Li,
A system of integral equations on half space, J. Math. Anal. Appl., 381 (2011), 392-401.
|
show all references
References:
[1] |
D. Applebeaum, Lévy Processes and Stochastic Calculus, second edition, Cambridge Studies in Advanced Mathematics, 116, Cambridge University Press, Cambridge, 2009. |
[2] |
J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972. |
[3] |
M. Belloni, V. Ferone and B. Kawohl,
Isoperimetric inequalities, Wulff shape and related questions for strongly nonlinear elliptic operators, Special issue dedicated to Lawrence E. Payne. Z. Angew. Math. Phys., 54 (2003), 771-783.
doi: 10.1007/s00033-003-3209-y. |
[4] |
J. Bertoin, Lévy Processes, Cambridge Tracts in Mathematics 121, Cambridge University Press, Cambridge, 1996. |
[5] |
P. J. Bouchard and A. Georges,
Anomalous diffusion in disordered media: Statistical mechanics, models and physical applications, Phys. Rep., 195 (1990), 127-293.
|
[6] |
C. Brandle, E. Colorado, A. de Pablo and U. Sanchez,
A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect., 143 (2013), 39-71.
doi: 10.1017/S0308210511000175. |
[7] |
X. Cabré and J. Tan,
Positive solutions of nonlinear problems involving the square root of the Laplacian, Advances in Math., 224 (2010), 2052-2053.
doi: 10.1016/j.aim.2010.01.025. |
[8] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Patial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[9] |
L. Caffarelli and L. Vasseur,
Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Ann. of Math., 171 (2010), 1903-1930.
doi: 10.4007/annals.2010.171.1903. |
[10] |
L. Cao and W. Chen,
Liouville type theorems for poly-harmonic Navier problems, Discrete Contin. Dyn. Syst., 33 (2013), 3937-3955.
doi: 10.3934/dcds.2013.33.3937. |
[11] |
W. Chen, C. Li and G. Li, Maximum principles for a fully nonlinear fractional order equation and symmetry of solutions, Calc. Var. Partial Differential Equations, 56 (2017).
doi: 10.1007/s00526-017-1110-3. |
[12] |
W. Chen, C. Li and Y. Li,
A direct method of moving planes for the fractional Laplacian, Advances in Math., 308 (2017), 404-437.
doi: 10.1016/j.aim.2016.11.038. |
[13] |
W. Chen, Y. Li and R. Zhang,
A direct method of moving spheres on fractional order equations, J. Funct. Anal., 272 (2017), 4131-4157.
doi: 10.1016/j.jfa.2017.02.022. |
[14] |
W. Chen and J. Zhu,
Radial symmetry and regularity of solutions for poly-harmonic Dirichlet problems, J. Math. Anal. Appl., 377 (2001), 744-753.
doi: 10.1016/j.jmaa.2010.11.035. |
[15] |
P. Constantin, Euler equations, Navier-Stokes equations and turbulence, Mathematical Foundation of Turbulent Viscous Flows, Lecture notes in Math. Springer, Berlin, 1871 (2006), 1–43.
doi: 10.1007/11545989_1. |
[16] |
M. Fall and T. Weth,
Nonexistence results for a class of fractional elliptic boundary value problems, J. Funct. Anal., 263 (2012), 2205-2227.
doi: 10.1016/j.jfa.2012.06.018. |
[17] |
S. Jarohs and T. Weth,
Symmetry via antisymmetric maximum principles in nonlocal problems of variable order, Ann. Mat. Pura Appl., 195 (2016), 273-291.
doi: 10.1007/s10231-014-0462-y. |
[18] |
D. Li, P. Niu and R. Zhuo,
Nonexistence of positive solutions for an integral equation related to the Hardy-Sobolev inequality, Acta Appl. Math., 134 (2014), 185-200.
doi: 10.1007/s10440-014-9878-z. |
[19] |
D. Li, P. Niu and R. Zhuo, Symmetry and non-existence of positive solutions for PDE system with Navier boundary conditions on a half space, Compex Var. Elliptic Equ., 59 (2014).
doi: 10.1080/17476933.2013.854346. |
[20] |
D. Li, P. Niu and R. Zhuo,
Symmetry and nonexistence of positive solutions of integral systems with hardy term, J. Math. Anal. Appl., 424 (2015), 915-931.
doi: 10.1016/j.jmaa.2014.11.029. |
[21] |
G. Lu and J. Zhu,
An overdetermined problem in Riesz-potential and fractional Laplacian, Nonlinear Anal., 75 (2012), 3036-3048.
doi: 10.1016/j.na.2011.11.036. |
[22] |
W. Reichel and H. Zou,
Non-existence results for semilinear cooperative elliptic systems via moving spheres, J. Differential Equations, 161 (2000), 219-243.
doi: 10.1006/jdeq.1999.3700. |
[23] |
V. Tarasov and G. Zaslasvky,
Fractional dynamics of systems with long-range interaction, Comm. Nonl Sci. Numer. Simul., 11 (2006), 885-898.
doi: 10.1016/j.cnsns.2006.03.005. |
[24] |
P. Wang and P. Niu,
A direct method of moving planes for a fully nonlinear nonlocal system, Commun. Pur. Appl. Anal., 16 (2017), 1707-1718.
|
[25] |
P. Wang and M. Yu,
Solutions of fully nonlinear nonlocal systems, J. Math. Anal. Appl., 450 (2017), 982-995.
|
[26] |
R. Zhuo, W. Chen, X. Cui and Z. Yuan,
Symmetry and non-existence of solutions for nonlinear system involving the fractional Laplacian, Discrete Contin. Dyn. Syst., 36 (2016), 1125-1141.
doi: 10.3934/dcds.2016.36.1125. |
[27] |
R. Zhuo and D. Li,
A system of integral equations on half space, J. Math. Anal. Appl., 381 (2011), 392-401.
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