Well-posedness results for fractional semi-linear wave equations

Jean-Daniel Djida; Arran Fernandez; Iván Area

doi:10.3934/dcdsb.2019255

Previous Article
Chaotic behavior in the unfolding of Hopf-Bogdanov-Takens singularities
DCDS-B Home
This Issue
Next Article
Existence of homoclinic solutions for a nonlinear fourth order $ p $-Laplacian difference equation

February 2020, 25(2): 569-597. doi: 10.3934/dcdsb.2019255

Well-posedness results for fractional semi-linear wave equations

Jean-Daniel Djida ^1, , Arran Fernandez ^2,, and Iván Area ^3,

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

^* Corresponding author: Arran Fernandez

Dedicated to Prof. Juan J. Nieto on the occasion of his 60th birthday

Received April 2019 Revised May 2019 Published February 2020 Early access November 2019

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.

Keywords: Fractional partial differential equations, fractional semi-linear wave equations, local and global existence of solutions, regularity estimates.

Mathematics Subject Classification: Primary: 26A33, 74G20, 74G25; Secondary: 35R11, 35G31, 35B65.

Citation: Jean-Daniel Djida, Arran Fernandez, Iván Area. Well-posedness results for fractional semi-linear wave equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 569-597. doi: 10.3934/dcdsb.2019255

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.
[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.
[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.

[1]	Li Ma, Lin Zhao. Regularity for positive weak solutions to semi-linear elliptic equations. Communications on Pure and Applied Analysis, 2008, 7 (3) : 631-643. doi: 10.3934/cpaa.2008.7.631
[2]	Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
[3]	Frank Pörner, Daniel Wachsmuth. Tikhonov regularization of optimal control problems governed by semi-linear partial differential equations. Mathematical Control and Related Fields, 2018, 8 (1) : 315-335. doi: 10.3934/mcrf.2018013
[4]	Wen Feng, Milena Stanislavova, Atanas Stefanov. On the spectral stability of ground states of semi-linear Schrödinger and Klein-Gordon equations with fractional dispersion. Communications on Pure and Applied Analysis, 2018, 17 (4) : 1371-1385. doi: 10.3934/cpaa.2018067
[5]	Yukang Chen, Changhua Wei. Partial regularity of solutions to the fractional Navier-Stokes equations. Discrete and Continuous Dynamical Systems, 2016, 36 (10) : 5309-5322. doi: 10.3934/dcds.2016033
[6]	Hantaek Bae. On the local and global existence of the Hall equations with fractional Laplacian and related equations. Networks and Heterogeneous Media, 2022 doi: 10.3934/nhm.2022021
[7]	Jason R. Morris. A Sobolev space approach for global solutions to certain semi-linear heat equations in bounded domains. Conference Publications, 2009, 2009 (Special) : 574-582. doi: 10.3934/proc.2009.2009.574
[8]	Nguyen Thieu Huy, Vu Thi Ngoc Ha, Pham Truong Xuan. Boundedness and stability of solutions to semi-linear equations and applications to fluid dynamics. Communications on Pure and Applied Analysis, 2016, 15 (6) : 2103-2116. doi: 10.3934/cpaa.2016029
[9]	Yongqin Liu. The point-wise estimates of solutions for semi-linear dissipative wave equation. Communications on Pure and Applied Analysis, 2013, 12 (1) : 237-252. doi: 10.3934/cpaa.2013.12.237
[10]	Priscila Santos Ramos, J. Vanterler da C. Sousa, E. Capelas de Oliveira. Existence and uniqueness of mild solutions for quasi-linear fractional integro-differential equations. Evolution Equations and Control Theory, 2022, 11 (1) : 1-24. doi: 10.3934/eect.2020100
[11]	Xiaoxue Ji, Pengcheng Niu, Pengyan Wang. Non-existence results for cooperative semi-linear fractional system via direct method of moving spheres. Communications on Pure and Applied Analysis, 2020, 19 (2) : 1111-1128. doi: 10.3934/cpaa.2020051
[12]	Rim Bourguiba, Rosana Rodríguez-López. Existence results for fractional differential equations in presence of upper and lower solutions. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1723-1747. doi: 10.3934/dcdsb.2020180
[13]	Abdon Atangana, Ali Akgül. On solutions of fractal fractional differential equations. Discrete and Continuous Dynamical Systems - S, 2021, 14 (10) : 3441-3457. doi: 10.3934/dcdss.2020421
[14]	Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure and Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053
[15]	Henri Schurz. Analysis and discretization of semi-linear stochastic wave equations with cubic nonlinearity and additive space-time noise. Discrete and Continuous Dynamical Systems - S, 2008, 1 (2) : 353-363. doi: 10.3934/dcdss.2008.1.353
[16]	Binjie Li, Xiaoping Xie. Regularity of solutions to time fractional diffusion equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 3195-3210. doi: 10.3934/dcdsb.2018340
[17]	Geng Chen, Yannan Shen. Existence and regularity of solutions in nonlinear wave equations. Discrete and Continuous Dynamical Systems, 2015, 35 (8) : 3327-3342. doi: 10.3934/dcds.2015.35.3327
[18]	Hua Chen, Nian Liu. Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equations with singular potentials. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 661-682. doi: 10.3934/dcds.2016.36.661
[19]	Jesus Idelfonso Díaz, Jean Michel Rakotoson. On very weak solutions of semi-linear elliptic equations in the framework of weighted spaces with respect to the distance to the boundary. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1037-1058. doi: 10.3934/dcds.2010.27.1037
[20]	Liping Luo, Zhenguo Luo, Yunhui Zeng. New results for oscillation of fractional partial differential equations with damping term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3223-3231. doi: 10.3934/dcdss.2020336

American Institute of Mathematical Sciences

Well-posedness results for fractional semi-linear wave equations

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Well-posedness results for fractional semi-linear wave equations

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors