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Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation
Well-posedness results for fractional semi-linear wave equations
1. | Departamento de Análise Matemática, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain |
2. | Department of Mathematics, Eastern Mediterranean University, Famagusta, Northern Cyprus, via Mersin-10, Turkey |
3. | Departamento de Matemática Aplicada II, E.E. Aeronáutica e do Espazo, Universidade de Vigo, 32004-Ourense, Spain |
This work is concerned with well-posedness results for nonlocal semi-linear wave equations involving the fractional Laplacian and fractional derivative operator taken in the sense of Caputo. Representations for solutions, existence of classical solutions, and some $ L^{p} $-estimates are derived, by considering a quasi-stationary elliptic problem that comes from the realisation of the fractional Laplacian as the Dirichlet-to-Neumann map for a non-uniformly elliptic problem posed on a semi-infinite cylinder. We derive some properties such as existence of global weak solutions of the extended semi-linear integro-differential equations.
References:
[1] |
N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, Contemporary Research in Elliptic PDEs and Related Topics, 2019, 1–105.
doi: 10.1007/978-3-030-18921-1_1. |
[2] |
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964. |
[3] |
M. Allen, L. Caffarelli and A. Vasseur,
A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603-630.
doi: 10.1007/s00205-016-0969-z. |
[4] |
E. Alvarez, C. G. Gal, V. Keyantuo and M. Warma,
Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Analysis, 181 (2019), 24-61.
doi: 10.1016/j.na.2018.10.016. |
[5] |
I. Athanasopoulos and L. A. Caffarelli,
Continuity of the temperature in boundary heat control problems, Adv. Math., 224 (2010), 293-315.
doi: 10.1016/j.aim.2009.11.010. |
[6] |
D. Baleanu and A. Fernandez,
A generalisation of the Malgrange-Ehrenpreis theorem to find fundamental solutions to fractional PDEs, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 1-12.
doi: 10.14232/ejqtde.2017.1.15. |
[7] |
A. Bernardis, F. J. Martín-Reyes, P. R. Stinga and J. L. Torrea,
Maximum principles, extension problem and inversion for nonlocal one-sided equations, J. Differential Equations, 260 (2016), 6333-6362.
doi: 10.1016/j.jde.2015.12.042. |
[8] |
M. Š. Birman and M. Z. Solomjak, Spektralnaya Teoriya Samosopryazhennykh Operatorov v Gilbertovom Prostranstve, Leningrad. Univ., Leningrad, 1980. Google Scholar |
[9] |
C. Bucur and F. Ferrari,
An extension problem for the fractional derivative defined by Marchaud, Fract. Calc. Appl. Anal., 19 (2016), 867-887.
doi: 10.1515/fca-2016-0047. |
[10] |
X. Cabré and J. G. 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 L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 2007, 1245–1260.
doi: 10.1080/03605300600987306. |
[12] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire,
Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[13] |
M. D'Abbicco, M. R. Ebert and T. Picon, Global existence of small data solutions to the semilinear fractional wave equation, New Trends in Analysis and Interdisciplinary Applications, Trends Math. Res. Perspect., Birkhäuser/Springer, Cham, (2017), 465–471.
doi: 10.1007/978-3-319-48812-7_59. |
[14] |
A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez,
A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.
doi: 10.1016/j.aim.2010.07.017. |
[15] |
A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez,
A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284.
doi: 10.1002/cpa.21408. |
[16] |
S. Dipierro and E. Valdinoci,
A simple mathematical model inspired by the Purkinje cells: From delayed travelling waves to fractional diffusion, Bull. Math. Biol., 80 (2018), 1849-1870.
doi: 10.1007/s11538-018-0437-z. |
[17] |
J.-D. Djida and A. Fernandez,
Interior regularity estimates for a degenerate elliptic equation with mixed boundary conditions, Axioms, 7 (2018), 1-16.
doi: 10.3390/axioms7030065. |
[18] |
S. D. Eidelman and A. N. Kochubei,
Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.
doi: 10.1016/j.jde.2003.12.002. |
[19] |
L. C. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[20] |
P. Felmer, A. Quaas and J. G. Tan,
Positive solutions of nonlinear schrödinger equation with the fractional laplacian, P. Roy. Soc. Edinb. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[21] |
A. Fernandez,
An elliptic regularity theorem for fractional partial differential operators, Comp. Appl. Math., 37 (2018), 5542-5553.
doi: 10.1007/s40314-018-0618-2. |
[22] |
Y. Fujita,
Energy inequalities for integro-partial differential equations with Riemann-Liouville integrals, SIAM J. Math. Anal., 23 (1992), 1182-118.
doi: 10.1137/0523066. |
[23] |
Y. V. Gorbatenko,
Existence and uniqueness of mild solutions of second order semilinear differential equations in Banach space, Methods Funct. Anal. Topology, 17 (2011), 1-9.
|
[24] |
R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, Heidelberg, 2014.
doi: 10.1007/978-3-662-43930-2. |
[25] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. |
[26] |
H. Hirata and C. X. Miao,
Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding to fractional-order time derivative., Adv. Differential Equations, 7 (2002), 217-236.
|
[27] |
Y. H. Huang,
Explicit Barenblatt profiles for fractional porous medium equations, Bulletin of the London Mathematical Society, 46 (2014), 857-869.
doi: 10.1112/blms/bdu045. |
[28] |
J. Kemppainen, J. Siljander and R. Zacher,
Representation of solutions and large-time behaviour for fully nonlocal diffusion equations, J. Differential Equations, 263 (2017), 149-201.
doi: 10.1016/j.jde.2017.02.030. |
[29] |
V. Keyantuo, C. Lizama and M. Warma, Existence, regularity and representation of solutions of time fractional wave equations, Electron. J. Differential Equations, 2017 (2017), 42 pp. |
[30] |
Y. Kian and M. Yamamoto,
On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. Appl. Anal., 20 (2017), 117-138.
doi: 10.1515/fca-2017-0006. |
[31] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. |
[32] |
K.-H. Kim and S. Lim,
Asymptotic behaviours of fundamental solution and its derivatives related to space-time fractional differential equations, J. Korean Math. Soc., 53 (2016), 929-967.
doi: 10.4134/JKMS.j150343. |
[33] |
A. N. Kochubeǐ,
Diffusion of fractional order, Differ. Equ., 26 (1990), 485-492.
|
[34] |
N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972. |
[35] |
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. III, Die Grundlehren der mathematischen Wissenschaften, Band 183. Springer-Verlag, New York-Heidelberg, 1973. |
[36] |
L. Maligranda,
On interpolation of nonlinear operators, Comment. Math. Prace Mat., 28 (1989), 253-275.
|
[37] |
L. Maligranda,
Marcinkiewicz interpolation theorem and marcinkiewicz spaces, Wiad. Mat., 48 (2012), 157-171.
doi: 10.14708/wm.v48i2.328. |
[38] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
![]() |
[39] |
B. Muckenhoupt,
Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226.
doi: 10.1090/S0002-9947-1972-0293384-6. |
[40] |
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. |
[41] |
A. Niang,
Boundary regularity for a degenerate elliptic equation with mixed boundary conditions, Commun. Pure Appl. Anal., 18 (2019), 107-128.
doi: 10.3934/cpaa.2019007. |
[42] |
R. H. Nochetto, E. Otárola and A. J. Salgado,
A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15 (2015), 733-791.
doi: 10.1007/s10208-014-9208-x. |
[43] |
E. Otárola and A. J. Salgado,
Regularity of solutions to space-time fractional wave equations: A PDE approach, Fract. Calc. Appl. Anal., 21 (2018), 1262-1293.
doi: 10.1515/fca-2018-0067. |
[44] |
A. Palmieri and M. Reissig,
Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation. II, Math. Nach., 291 (2018), 1859-1892.
doi: 10.1002/mana.201700144. |
[45] |
K. Sakamoto and M. Yamamoto,
Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.
doi: 10.1016/j.jmaa.2011.04.058. |
[46] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. |
[47] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[48] |
P. R. Stinga and J. L. Torrea,
Extension problem and Harnack's inequality for some fractional operators, Comm. Part. Diff. Eqs., 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[49] |
L. Tartar, An introduction to Sobolev spaces and interpolation spaces., Lecture Notes of the Unione Matematica Italiana, volume 3, Springer, Berlin, 2007. |
[50] |
B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, 1736. Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103908. |
[51] |
E. Valdinoci,
From the lung jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.
|
[52] |
J. L. Vázquez and B. Volzone,
Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type, J. Math. Pures Appl., 101 (2014), 553-582.
doi: 10.1016/j.matpur.2013.07.001. |
[53] |
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995.
![]() |
show all references
References:
[1] |
N. Abatangelo and E. Valdinoci, Getting acquainted with the fractional Laplacian, Contemporary Research in Elliptic PDEs and Related Topics, 2019, 1–105.
doi: 10.1007/978-3-030-18921-1_1. |
[2] |
M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Applied Mathematics Series, 55 For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 1964. |
[3] |
M. Allen, L. Caffarelli and A. Vasseur,
A parabolic problem with a fractional time derivative, Arch. Ration. Mech. Anal., 221 (2016), 603-630.
doi: 10.1007/s00205-016-0969-z. |
[4] |
E. Alvarez, C. G. Gal, V. Keyantuo and M. Warma,
Well-posedness results for a class of semi-linear super-diffusive equations, Nonlinear Analysis, 181 (2019), 24-61.
doi: 10.1016/j.na.2018.10.016. |
[5] |
I. Athanasopoulos and L. A. Caffarelli,
Continuity of the temperature in boundary heat control problems, Adv. Math., 224 (2010), 293-315.
doi: 10.1016/j.aim.2009.11.010. |
[6] |
D. Baleanu and A. Fernandez,
A generalisation of the Malgrange-Ehrenpreis theorem to find fundamental solutions to fractional PDEs, Electron. J. Qual. Theory Differ. Equ., 2017 (2017), 1-12.
doi: 10.14232/ejqtde.2017.1.15. |
[7] |
A. Bernardis, F. J. Martín-Reyes, P. R. Stinga and J. L. Torrea,
Maximum principles, extension problem and inversion for nonlocal one-sided equations, J. Differential Equations, 260 (2016), 6333-6362.
doi: 10.1016/j.jde.2015.12.042. |
[8] |
M. Š. Birman and M. Z. Solomjak, Spektralnaya Teoriya Samosopryazhennykh Operatorov v Gilbertovom Prostranstve, Leningrad. Univ., Leningrad, 1980. Google Scholar |
[9] |
C. Bucur and F. Ferrari,
An extension problem for the fractional derivative defined by Marchaud, Fract. Calc. Appl. Anal., 19 (2016), 867-887.
doi: 10.1515/fca-2016-0047. |
[10] |
X. Cabré and J. G. 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 L. Silvestre, An extension problem related to the fractional Laplacian, Comm. Part. Diff. Eq., 32 2007, 1245–1260.
doi: 10.1080/03605300600987306. |
[12] |
A. Capella, J. Dávila, L. Dupaigne and Y. Sire,
Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), 1353-1384.
doi: 10.1080/03605302.2011.562954. |
[13] |
M. D'Abbicco, M. R. Ebert and T. Picon, Global existence of small data solutions to the semilinear fractional wave equation, New Trends in Analysis and Interdisciplinary Applications, Trends Math. Res. Perspect., Birkhäuser/Springer, Cham, (2017), 465–471.
doi: 10.1007/978-3-319-48812-7_59. |
[14] |
A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez,
A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409.
doi: 10.1016/j.aim.2010.07.017. |
[15] |
A. de Pablo, F. Quirós, A. Rodríguez and J. L. Vázquez,
A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284.
doi: 10.1002/cpa.21408. |
[16] |
S. Dipierro and E. Valdinoci,
A simple mathematical model inspired by the Purkinje cells: From delayed travelling waves to fractional diffusion, Bull. Math. Biol., 80 (2018), 1849-1870.
doi: 10.1007/s11538-018-0437-z. |
[17] |
J.-D. Djida and A. Fernandez,
Interior regularity estimates for a degenerate elliptic equation with mixed boundary conditions, Axioms, 7 (2018), 1-16.
doi: 10.3390/axioms7030065. |
[18] |
S. D. Eidelman and A. N. Kochubei,
Cauchy problem for fractional diffusion equations, J. Differential Equations, 199 (2004), 211-255.
doi: 10.1016/j.jde.2003.12.002. |
[19] |
L. C. Evans, Partial Differential Equations, Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/gsm/019. |
[20] |
P. Felmer, A. Quaas and J. G. Tan,
Positive solutions of nonlinear schrödinger equation with the fractional laplacian, P. Roy. Soc. Edinb. A, 142 (2012), 1237-1262.
doi: 10.1017/S0308210511000746. |
[21] |
A. Fernandez,
An elliptic regularity theorem for fractional partial differential operators, Comp. Appl. Math., 37 (2018), 5542-5553.
doi: 10.1007/s40314-018-0618-2. |
[22] |
Y. Fujita,
Energy inequalities for integro-partial differential equations with Riemann-Liouville integrals, SIAM J. Math. Anal., 23 (1992), 1182-118.
doi: 10.1137/0523066. |
[23] |
Y. V. Gorbatenko,
Existence and uniqueness of mild solutions of second order semilinear differential equations in Banach space, Methods Funct. Anal. Topology, 17 (2011), 1-9.
|
[24] |
R. Gorenflo, A. A. Kilbas, F. Mainardi and S. V. Rogosin, Mittag-Leffler Functions, Related Topics and Applications, Springer Monographs in Mathematics. Springer, Heidelberg, 2014.
doi: 10.1007/978-3-662-43930-2. |
[25] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840. Springer-Verlag, Berlin-New York, 1981. |
[26] |
H. Hirata and C. X. Miao,
Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding to fractional-order time derivative., Adv. Differential Equations, 7 (2002), 217-236.
|
[27] |
Y. H. Huang,
Explicit Barenblatt profiles for fractional porous medium equations, Bulletin of the London Mathematical Society, 46 (2014), 857-869.
doi: 10.1112/blms/bdu045. |
[28] |
J. Kemppainen, J. Siljander and R. Zacher,
Representation of solutions and large-time behaviour for fully nonlocal diffusion equations, J. Differential Equations, 263 (2017), 149-201.
doi: 10.1016/j.jde.2017.02.030. |
[29] |
V. Keyantuo, C. Lizama and M. Warma, Existence, regularity and representation of solutions of time fractional wave equations, Electron. J. Differential Equations, 2017 (2017), 42 pp. |
[30] |
Y. Kian and M. Yamamoto,
On existence and uniqueness of solutions for semilinear fractional wave equations, Fract. Calc. Appl. Anal., 20 (2017), 117-138.
doi: 10.1515/fca-2017-0006. |
[31] |
A. A. Kilbas, H. M. Srivastava and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematics Studies, 204. Elsevier Science B.V., Amsterdam, 2006. |
[32] |
K.-H. Kim and S. Lim,
Asymptotic behaviours of fundamental solution and its derivatives related to space-time fractional differential equations, J. Korean Math. Soc., 53 (2016), 929-967.
doi: 10.4134/JKMS.j150343. |
[33] |
A. N. Kochubeǐ,
Diffusion of fractional order, Differ. Equ., 26 (1990), 485-492.
|
[34] |
N. S. Landkof, Foundations of Modern Potential Theory, Die Grundlehren der mathematischen Wissenschaften, Band 180. Springer-Verlag, New York-Heidelberg, 1972. |
[35] |
J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. III, Die Grundlehren der mathematischen Wissenschaften, Band 183. Springer-Verlag, New York-Heidelberg, 1973. |
[36] |
L. Maligranda,
On interpolation of nonlinear operators, Comment. Math. Prace Mat., 28 (1989), 253-275.
|
[37] |
L. Maligranda,
Marcinkiewicz interpolation theorem and marcinkiewicz spaces, Wiad. Mat., 48 (2012), 157-171.
doi: 10.14708/wm.v48i2.328. |
[38] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
![]() |
[39] |
B. Muckenhoupt,
Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226.
doi: 10.1090/S0002-9947-1972-0293384-6. |
[40] |
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. |
[41] |
A. Niang,
Boundary regularity for a degenerate elliptic equation with mixed boundary conditions, Commun. Pure Appl. Anal., 18 (2019), 107-128.
doi: 10.3934/cpaa.2019007. |
[42] |
R. H. Nochetto, E. Otárola and A. J. Salgado,
A PDE approach to fractional diffusion in general domains: A priori error analysis, Found. Comput. Math., 15 (2015), 733-791.
doi: 10.1007/s10208-014-9208-x. |
[43] |
E. Otárola and A. J. Salgado,
Regularity of solutions to space-time fractional wave equations: A PDE approach, Fract. Calc. Appl. Anal., 21 (2018), 1262-1293.
doi: 10.1515/fca-2018-0067. |
[44] |
A. Palmieri and M. Reissig,
Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation. II, Math. Nach., 291 (2018), 1859-1892.
doi: 10.1002/mana.201700144. |
[45] |
K. Sakamoto and M. Yamamoto,
Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl., 382 (2011), 426-447.
doi: 10.1016/j.jmaa.2011.04.058. |
[46] |
S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. |
[47] |
L. Silvestre,
Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007), 67-112.
doi: 10.1002/cpa.20153. |
[48] |
P. R. Stinga and J. L. Torrea,
Extension problem and Harnack's inequality for some fractional operators, Comm. Part. Diff. Eqs., 35 (2010), 2092-2122.
doi: 10.1080/03605301003735680. |
[49] |
L. Tartar, An introduction to Sobolev spaces and interpolation spaces., Lecture Notes of the Unione Matematica Italiana, volume 3, Springer, Berlin, 2007. |
[50] |
B. O. Turesson, Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Mathematics, 1736. Springer-Verlag, Berlin, 2000.
doi: 10.1007/BFb0103908. |
[51] |
E. Valdinoci,
From the lung jump random walk to the fractional Laplacian, Bol. Soc. Esp. Mat. Apl., 49 (2009), 33-44.
|
[52] |
J. L. Vázquez and B. Volzone,
Symmetrization for linear and nonlinear fractional parabolic equations of porous medium type, J. Math. Pures Appl., 101 (2014), 553-582.
doi: 10.1016/j.matpur.2013.07.001. |
[53] |
G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995.
![]() |
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