doi: 10.3934/dcdsb.2019049

Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation

1. 

Departamento de Estatística, Análise Matemática e Optimización, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

2. 

Departamento de Estatística, Análise Matemática e Optimización, Instituto de Matemáticas, Universidade de Santiago de Compostela, 15782 Santiago de Compostela, Spain

3. 

Departamento de Matemática Aplicada Ⅱ, E.E. Aeronáutica e do Espazo, Universidade de Vigo, Campus As Lagoas s/n, 32004 Vigo, Spain

* Corresponding author: jeandaniel.djida@usc.es

Received  March 2018 Revised  September 2018 Published  February 2019

Fund Project: This work has been partially supported by the Agencia Estatal de Investigación (AEI) of Spain under grant MTM2016–75140–P, co-financed by the European Community fund FEDER, and Xunta de Galicia, grants GRC 2015–004 and R 2016/022

We study a fractional time porous medium equation with fractional potential pressure. The initial data is assumed to be a bounded function with compact support and fast decay at infinity. We establish existence of weak solutions for which we determine whether the property of compact support is conserved in time depending on some parameters of the problem. Special attention is paid to the property of finite propagation for specific values of the parameters.

Citation: Jean-Daniel Djida, Juan J. Nieto, Iván Area. Nonlocal time-porous medium equation: Weak solutions and finite speed of propagation. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2019049
References:
[1]

M. Allen, Hölder regularity for nondivergence nonlocal parabolic equations, Calc. Var. Partial Differential Equations, 57 (2018), Art. 110, 29 pp, arXiv: 1610.10073 doi: 10.1007/s00526-018-1367-1.

[2]

M. Allen, A nondivergence parabolic problem with a fractional time derivative, Differential Integral Equations, 31 (2018), 215–230.

[3]

M. AllenL. 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]

M. AllenL. Caffarelli and A. Vasseur, Porous medium flow with both a fractional potential pressure and fractional time derivative, Chin. Ann. Math. Ser. B, 38 (2017), 45-82. doi: 10.1007/s11401-016-1063-4.

[5]

I. AreaJ. Losada and J. J. Nieto, A note on the fractional logistic equation, Phys. A, 444 (2016), 182-187. doi: 10.1016/j.physa.2015.10.037.

[6]

D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291.

[7]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. doi: 10.2298/TSCI160111018A.

[8]

D. Baleanu and A. Fernandez, A generalisation of the malgrange-ehrenpreis theorem to find fundamental solutions to fractional PDEs, Electronic Journal of Qualitative Theory of Differential Equations, 2017 (2017), 1-12. doi: 10.14232/ejqtde.2017.1.15.

[9]

D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 444–462. doi: 10.1016/j.cnsns.2017.12.003.

[10]

A. BernardisF. J. Martín-ReyesP. 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.

[11]

P. Biler, C. Imbert and G. Karch, The nonlocal porous medium equation: Barenblatt profiles and other weak solutions, Archive for Rational Mechanics and Analysis, 215 (2015), 497–529. doi: 10.1007/s00205-014-0786-1.

[12]

P. BilerG. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions, Commun. Math. Phys., 294 (2010), 145-168. doi: 10.1007/s00220-009-0855-8.

[13]

M. BonforteA. Figalli and X. Ros-Oton, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains, Comm. Pure Appl. Math., 70 (2017), 1472-1508. doi: 10.1002/cpa.21673.

[14]

M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Adv. Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018.

[15]

C. Bucur, Some nonlocal operators and effects due to nonlocality, preprint, arXiv: 1705.00953

[16]

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.

[17]

L. CaffarelliF. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow, Journal Europ. Math. Society., 15 (2013), 1701-1746. doi: 10.4171/JEMS/401.

[18]

L. Caffarelli and J. L. Vázquez, Asymptotic behaviour of a porous medium equation with fractional diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 1393-1404. doi: 10.3934/dcds.2011.29.1393.

[19]

L. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure, Arch. Ration. Mech. Anal., 202 (2011), 537-565. doi: 10.1007/s00205-011-0420-4.

[20]

M. Caputo, Diffusion of fluids in porous media with memory, Geothermics, 28 (1999), 113-130. doi: 10.1016/S0375-6505(98)00047-9.

[21]

M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr Fract Differ Appl, 1 (2015), 73-85.

[22]

J. A. CarrilloY. HuangM. C. Santos and J. L. Vázquez, Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure, J. Differ. Equations, 258 (2015), 736-763. doi: 10.1016/j.jde.2014.10.003.

[23]

J. D. DjidaA. Atangana and I. Area, Numerical computation of a fractional derivative with non-local and non-singular kernel, Math. Model. Nat. Phenom., 12 (2017), 4-13. doi: 10.1051/mmnp/201712302.

[24]

J. D. Djida, J. J. Nieto and I. Area, Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel, Discrete Continuous Dyn. Syst. Ser. S (to appear), 2018.

[25]

R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Masson, Paris, 1987.

[26]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017.

[27]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284. doi: 10.1002/cpa.21408.

[28]

E. Di NezzaG. 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.

[29]

S. Dipierro, E. Valdinoci and V. Vespri, Decay estimates for evolutionary equations with fractional time diffusion, preprint, arXiv: 1707.08278v1.

[30]

F. Ferrari, Weyl and Marchaud derivatives: A forgotten history, Mathematics, 6 (2018), p6. doi: 10.3390/math6010006.

[31]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Berlin: Springer, reprint of the 1998 ed. edition, 2001.

[32]

J. KemppainenJ. Siljander and R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, Journal of Differential Equations, 263 (2017), 149-201. doi: 10.1016/j.jde.2017.02.030.

[33]

P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Communications on Pure and Applied Analysis, 13 (2014), 2543-2557. doi: 10.3934/cpaa.2014.13.2543.

[34]

A. N. Kochubei, Fractional-order diffusion, Differ. Equations, 26 (1990), 485-492.

[35]

L. LiuT. Caraballo and P. E. Kloeden, Long time behavior of stochastic parabolic problems with white noise in materials with thermal memory, Revista Matemática Complutense, 30 (2018), 687-717. doi: 10.1007/s13163-017-0238-1.

[36]

F. MainardiY. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153-192.

[37]

F. Mainardi and G. Pagnini, The Wright functions as solutions of the time-fractional diffusion equation, Applied Mathematics and Computation, 141 (2003), 51-62. doi: 10.1016/S0096-3003(02)00320-X.

[38]

A. Marchaud, Sur Les Dérivées et Sur les Différences des Fonctions De Variables Réelles, PhD thesis, Faculté des Sciences de Paris, 1927.

[39]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

[40]

G. M. Mophou and G. M. N'Guérékata, On a class of fractional differential equations in a Sobolev space, Applicable Analysis, 91 (2012), 15-34. doi: 10.1080/00036811.2010.534730.

[41]

M. K. Saad, A. Atangana and D. Baleanu, New fractional derivatives with non-singular kernel applied to the Burgers equation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284.

[42]

S. Samko, A. A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Taylor & Francis, 1993.

[43]

J. Simon, Compact sets in the space $ L^p(0, t; b) $, Annali di Matematica Pura ed Applicata, 146 (1987), 65–96. doi: 10.1007/BF01762360.

[44]

D. StanF. del Teso and J. L. Vázquez, Finite and infinite speed of propagation for porous medium equations with fractional pressure, Comptes Rendus Acad. Sci., 352 (2014), 123-128. doi: 10.1016/j.crma.2013.12.003.

[45] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[46] J. L. Vázquez, The Porous Medium Equation, Oxford University Press, 2007.
[47]

J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, Nonlinear Partial Differential Equations, 271–298, Abel Symp., 7, Springer, Heidelberg, 2012. Available online at http://www.uam.es/personal_pdi/ciencias/jvazquez/JLV ABEL-2010.pdf. doi: 10.1007/978-3-642-25361-4_15.

show all references

References:
[1]

M. Allen, Hölder regularity for nondivergence nonlocal parabolic equations, Calc. Var. Partial Differential Equations, 57 (2018), Art. 110, 29 pp, arXiv: 1610.10073 doi: 10.1007/s00526-018-1367-1.

[2]

M. Allen, A nondivergence parabolic problem with a fractional time derivative, Differential Integral Equations, 31 (2018), 215–230.

[3]

M. AllenL. 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]

M. AllenL. Caffarelli and A. Vasseur, Porous medium flow with both a fractional potential pressure and fractional time derivative, Chin. Ann. Math. Ser. B, 38 (2017), 45-82. doi: 10.1007/s11401-016-1063-4.

[5]

I. AreaJ. Losada and J. J. Nieto, A note on the fractional logistic equation, Phys. A, 444 (2016), 182-187. doi: 10.1016/j.physa.2015.10.037.

[6]

D. G. Aronson and J. Serrin, Local behavior of solutions of quasilinear parabolic equations, Arch. Ration. Mech. Anal., 25 (1967), 81-122. doi: 10.1007/BF00281291.

[7]

A. Atangana and D. Baleanu, New fractional derivatives with nonlocal and non-singular kernel: Theory and application to heat transfer model, Thermal Science, 20 (2016), 763-769. doi: 10.2298/TSCI160111018A.

[8]

D. Baleanu and A. Fernandez, A generalisation of the malgrange-ehrenpreis theorem to find fundamental solutions to fractional PDEs, Electronic Journal of Qualitative Theory of Differential Equations, 2017 (2017), 1-12. doi: 10.14232/ejqtde.2017.1.15.

[9]

D. Baleanu and A. Fernandez, On some new properties of fractional derivatives with Mittag-Leffler kernel, Communications in Nonlinear Science and Numerical Simulation, 59 (2018), 444–462. doi: 10.1016/j.cnsns.2017.12.003.

[10]

A. BernardisF. J. Martín-ReyesP. 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.

[11]

P. Biler, C. Imbert and G. Karch, The nonlocal porous medium equation: Barenblatt profiles and other weak solutions, Archive for Rational Mechanics and Analysis, 215 (2015), 497–529. doi: 10.1007/s00205-014-0786-1.

[12]

P. BilerG. Karch and R. Monneau, Nonlinear diffusion of dislocation density and self-similar solutions, Commun. Math. Phys., 294 (2010), 145-168. doi: 10.1007/s00220-009-0855-8.

[13]

M. BonforteA. Figalli and X. Ros-Oton, Infinite speed of propagation and regularity of solutions to the fractional porous medium equation in general domains, Comm. Pure Appl. Math., 70 (2017), 1472-1508. doi: 10.1002/cpa.21673.

[14]

M. Bonforte and J. L. Vázquez, Quantitative local and global a priori estimates for fractional nonlinear diffusion equations, Adv. Math., 250 (2014), 242-284. doi: 10.1016/j.aim.2013.09.018.

[15]

C. Bucur, Some nonlocal operators and effects due to nonlocality, preprint, arXiv: 1705.00953

[16]

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.

[17]

L. CaffarelliF. Soria and J. L. Vázquez, Regularity of solutions of the fractional porous medium flow, Journal Europ. Math. Society., 15 (2013), 1701-1746. doi: 10.4171/JEMS/401.

[18]

L. Caffarelli and J. L. Vázquez, Asymptotic behaviour of a porous medium equation with fractional diffusion, Discrete Contin. Dyn. Syst., 29 (2011), 1393-1404. doi: 10.3934/dcds.2011.29.1393.

[19]

L. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractional potential pressure, Arch. Ration. Mech. Anal., 202 (2011), 537-565. doi: 10.1007/s00205-011-0420-4.

[20]

M. Caputo, Diffusion of fluids in porous media with memory, Geothermics, 28 (1999), 113-130. doi: 10.1016/S0375-6505(98)00047-9.

[21]

M. Caputo and M. Fabrizio, A new Definition of Fractional Derivative without Singular Kernel, Progr Fract Differ Appl, 1 (2015), 73-85.

[22]

J. A. CarrilloY. HuangM. C. Santos and J. L. Vázquez, Exponential convergence towards stationary states for the 1D porous medium equation with fractional pressure, J. Differ. Equations, 258 (2015), 736-763. doi: 10.1016/j.jde.2014.10.003.

[23]

J. D. DjidaA. Atangana and I. Area, Numerical computation of a fractional derivative with non-local and non-singular kernel, Math. Model. Nat. Phenom., 12 (2017), 4-13. doi: 10.1051/mmnp/201712302.

[24]

J. D. Djida, J. J. Nieto and I. Area, Parabolic problem with fractional time derivative with nonlocal and nonsingular Mittag-Leffler kernel, Discrete Continuous Dyn. Syst. Ser. S (to appear), 2018.

[25]

R. Dautray and J.-L. Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques, Masson, Paris, 1987.

[26]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A fractional porous medium equation, Adv. Math., 226 (2011), 1378-1409. doi: 10.1016/j.aim.2010.07.017.

[27]

A. de PabloF. QuirósA. Rodríguez and J. L. Vázquez, A general fractional porous medium equation, Comm. Pure Appl. Math., 65 (2012), 1242-1284. doi: 10.1002/cpa.21408.

[28]

E. Di NezzaG. 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.

[29]

S. Dipierro, E. Valdinoci and V. Vespri, Decay estimates for evolutionary equations with fractional time diffusion, preprint, arXiv: 1707.08278v1.

[30]

F. Ferrari, Weyl and Marchaud derivatives: A forgotten history, Mathematics, 6 (2018), p6. doi: 10.3390/math6010006.

[31]

D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations Of Second Order, Berlin: Springer, reprint of the 1998 ed. edition, 2001.

[32]

J. KemppainenJ. Siljander and R. Zacher, Representation of solutions and large-time behavior for fully nonlocal diffusion equations, Journal of Differential Equations, 263 (2017), 149-201. doi: 10.1016/j.jde.2017.02.030.

[33]

P. E. Kloeden and J. Simsen, Pullback attractors for non-autonomous evolution equations with spatially variable exponents, Communications on Pure and Applied Analysis, 13 (2014), 2543-2557. doi: 10.3934/cpaa.2014.13.2543.

[34]

A. N. Kochubei, Fractional-order diffusion, Differ. Equations, 26 (1990), 485-492.

[35]

L. LiuT. Caraballo and P. E. Kloeden, Long time behavior of stochastic parabolic problems with white noise in materials with thermal memory, Revista Matemática Complutense, 30 (2018), 687-717. doi: 10.1007/s13163-017-0238-1.

[36]

F. MainardiY. Luchko and G. Pagnini, The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4 (2001), 153-192.

[37]

F. Mainardi and G. Pagnini, The Wright functions as solutions of the time-fractional diffusion equation, Applied Mathematics and Computation, 141 (2003), 51-62. doi: 10.1016/S0096-3003(02)00320-X.

[38]

A. Marchaud, Sur Les Dérivées et Sur les Différences des Fonctions De Variables Réelles, PhD thesis, Faculté des Sciences de Paris, 1927.

[39]

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley, New York, 1993.

[40]

G. M. Mophou and G. M. N'Guérékata, On a class of fractional differential equations in a Sobolev space, Applicable Analysis, 91 (2012), 15-34. doi: 10.1080/00036811.2010.534730.

[41]

M. K. Saad, A. Atangana and D. Baleanu, New fractional derivatives with non-singular kernel applied to the Burgers equation, Chaos: An Interdisciplinary Journal of Nonlinear Science, 28 (2018), 063109, 6 pp. doi: 10.1063/1.5026284.

[42]

S. Samko, A. A. Kilbas and O. Marichev, Fractional Integrals and Derivatives, Taylor & Francis, 1993.

[43]

J. Simon, Compact sets in the space $ L^p(0, t; b) $, Annali di Matematica Pura ed Applicata, 146 (1987), 65–96. doi: 10.1007/BF01762360.

[44]

D. StanF. del Teso and J. L. Vázquez, Finite and infinite speed of propagation for porous medium equations with fractional pressure, Comptes Rendus Acad. Sci., 352 (2014), 123-128. doi: 10.1016/j.crma.2013.12.003.

[45] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
[46] J. L. Vázquez, The Porous Medium Equation, Oxford University Press, 2007.
[47]

J. L. Vázquez, Nonlinear diffusion with fractional laplacian operators, Nonlinear Partial Differential Equations, 271–298, Abel Symp., 7, Springer, Heidelberg, 2012. Available online at http://www.uam.es/personal_pdi/ciencias/jvazquez/JLV ABEL-2010.pdf. doi: 10.1007/978-3-642-25361-4_15.

[1]

Ran Zhuo, Wenxiong Chen, Xuewei Cui, Zixia Yuan. Symmetry and non-existence of solutions for a nonlinear system involving the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2016, 36 (2) : 1125-1141. doi: 10.3934/dcds.2016.36.1125

[2]

Eric R. Kaufmann. Existence and nonexistence of positive solutions for a nonlinear fractional boundary value problem. Conference Publications, 2009, 2009 (Special) : 416-423. doi: 10.3934/proc.2009.2009.416

[3]

Juan-Luis Vázquez. Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 857-885. doi: 10.3934/dcdss.2014.7.857

[4]

Lishan Lin. A priori bounds and existence result of positive solutions for fractional Laplacian systems. Discrete & Continuous Dynamical Systems - A, 2019, 39 (3) : 1517-1531. doi: 10.3934/dcds.2019065

[5]

Dengfeng Lü, Shuangjie Peng. On the positive vector solutions for nonlinear fractional Laplacian systems with linear coupling. Discrete & Continuous Dynamical Systems - A, 2017, 37 (6) : 3327-3352. doi: 10.3934/dcds.2017141

[6]

Tadeusz Kulczycki, Robert Stańczy. Multiple solutions for Dirichlet nonlinear BVPs involving fractional Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2581-2591. doi: 10.3934/dcdsb.2014.19.2581

[7]

Begoña Barrios, Leandro Del Pezzo, Jorge García-Melián, Alexander Quaas. A Liouville theorem for indefinite fractional diffusion equations and its application to existence of solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5731-5746. doi: 10.3934/dcds.2017248

[8]

Hao Yang, Fuke Wu, Peter E. Kloeden. Existence and approximation of strong solutions of SDEs with fractional diffusion coefficients. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-15. doi: 10.3934/dcdsb.2019071

[9]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 709-722. doi: 10.3934/dcdss.2020039

[10]

De Tang, Yanqin Fang. Regularity and nonexistence of solutions for a system involving the fractional Laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2431-2451. doi: 10.3934/cpaa.2015.14.2431

[11]

Lizhi Zhang. Symmetry of solutions to semilinear equations involving the fractional laplacian. Communications on Pure & Applied Analysis, 2015, 14 (6) : 2393-2409. doi: 10.3934/cpaa.2015.14.2393

[12]

Tingzhi Cheng. Monotonicity and symmetry of solutions to fractional Laplacian equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3587-3599. doi: 10.3934/dcds.2017154

[13]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615

[14]

Wanwan Wang, Hongxia Zhang, Huyuan Chen. Remarks on weak solutions of fractional elliptic equations. Communications on Pure & Applied Analysis, 2016, 15 (2) : 335-340. doi: 10.3934/cpaa.2016.15.335

[15]

Yutong Chen, Jiabao Su. Resonant problems for fractional Laplacian. Communications on Pure & Applied Analysis, 2017, 16 (1) : 163-188. doi: 10.3934/cpaa.2017008

[16]

Eugenio Montefusco, Benedetta Pellacci, Gianmaria Verzini. Fractional diffusion with Neumann boundary conditions: The logistic equation. Discrete & Continuous Dynamical Systems - B, 2013, 18 (8) : 2175-2202. doi: 10.3934/dcdsb.2013.18.2175

[17]

Alexander Quaas, Aliang Xia. Existence and uniqueness of positive solutions for a class of logistic type elliptic equations in $\mathbb{R}^N$ involving fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2653-2668. doi: 10.3934/dcds.2017113

[18]

Nicola Abatangelo. Large $s$-harmonic functions and boundary blow-up solutions for the fractional Laplacian. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5555-5607. doi: 10.3934/dcds.2015.35.5555

[19]

Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete & Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483

[20]

Miaomiao Niu, Zhongwei Tang. Least energy solutions for nonlinear Schrödinger equation involving the fractional Laplacian and critical growth. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 3963-3987. doi: 10.3934/dcds.2017168

2017 Impact Factor: 0.972

Metrics

  • PDF downloads (53)
  • HTML views (280)
  • Cited by (0)

Other articles
by authors

[Back to Top]