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March 2019, 24(3): 1297-1307. doi: 10.3934/dcdsb.2019017

Existence of solutions for space-fractional parabolic hemivariational inequalities

1. 

Guangxi Colleges and Universities Key Laboratory of Complex System Optimization, and Big Data Processing, Yulin Normal University, Yulin 537000, China

2. 

Guangxi Colleges and Universities Key Laboratory of Optimization Control, and Engineering Calculation, and College of Sciences, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

3. 

Center for Fundamental Science, and Research Center for Nonlinear Analysis and Optimization, Kaohsiung Medical University, Kaohsiung, 80708, Taiwan

4. 

Department of Medical Research, Kaohsiung Medical University Hospital, Kaohsiung, 80708, Taiwan

* Corresponding authors: zhhliu@hotmail.com(Z.Liu) and cfwen@kmu.edu.tw(C.Wen)

Received  October 2017 Revised  March 2018 Published  January 2019

Fund Project: The work was supported by NNSF of China (No.11671101), NSF of Guangxi (No.2018GXNSFDA138002), the High Level Innovation Team Program from Guangxi Higher Education Institutions of China (No. [2018] 35). This project has received funding from the European Unions Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie Grant Agreement No.823731 CONMECH

This paper is devoted to the existence of solutions for space-fractional parabolic hemivariational inequalities by means of the well-known surjectivity result for multivalued ($S_+$) type mappings.

Citation: Yongjian Liu, Zhenhai Liu, Ching-Feng Wen. Existence of solutions for space-fractional parabolic hemivariational inequalities. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1297-1307. doi: 10.3934/dcdsb.2019017
References:
[1]

G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159.

[2]

X. Cabre and J. Sola-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.

[3]

L. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119.

[4]

L. CaffarelliJ. M. Roquejoffre and Y. Sire, Variational problems in free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.

[5]

L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Mathematics. Second Series, 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.

[6]

K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129. doi: 10.1016/0022-247X(81)90095-0.

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

[8]

N. Costea and C. Varga, Systems of nonlinear hemivariational inequalities and applications, Topological Methods in Nonlinear Analysis, 41 (2013), 39-65.

[9]

W. Craig and M. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19 (1994), 367-389. doi: 10.1016/0165-2125(94)90003-5.

[10]

W. CraigU. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincare, Anal. Non Lineaire, 14 (1997), 615-667. doi: 10.1016/S0294-1449(97)80128-X.

[11]

L. Gasinski and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman and Hall/CRC, Boca Raton, London, New York, Washington, DC, 2005.

[12]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer, Dordrecht, The Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.

[13]

Z. H. Liu, A class of evolution hamivariational inequalities, Nonlinear Analysis, TMA, 36 (1999), 91-100. doi: 10.1016/S0362-546X(98)00016-9.

[14]

Z. H. Liu, Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities, Inverse Problems, 21 (2005), 13-20. doi: 10.1088/0266-5611/21/1/002.

[15]

Z. H. Liu, Existence results for quasilinear parabolic hemivariational inequalities, J. Differential Equations, 244 (2008), 1395-1409. doi: 10.1016/j.jde.2007.09.001.

[16]

Z. H. LiuX. W. Li and D. Motreanu, Approximate Controllability for Nonlinear Evolution Hemivariational Inequalities in Hilbert Spaces, SIAM Journal on Control and Optimization, 53 (2015), 3228-3244. doi: 10.1137/140994058.

[17]

Z. H. Liu and X. W. Li, Approximate controllability for a class of hemivariational inequalities, Nonlinear Analysis: Real World Applications, 22 (2015), 581-591. doi: 10.1016/j.nonrwa.2014.08.010.

[18]

Z. H. Liu, Anti-periodic solutions to nonlinear evolution equations, Journal of functional analysis, 258 (2010), 2026-2033. doi: 10.1016/j.jfa.2009.11.018.

[19]

Z. H. Liu and J. Tan, Nonlocal Elliptic Hemivariational inequalities, Electronic Journal of Qualitative Theory of Differential Equations, 66 (2017), 1-7. doi: 10.14232/ejqtde.2017.1.66.

[20]

Z. H. Liu and S. S. Zhang, On the degree theory for multivalued ($S_+$) type mappings, Applied Mathematics and Mechanics, 19 (1998), 1141-1149. doi: 10.1007/BF02456635.

[21]

A. Majda and E. Tabak, A two-dimensional model for quasigeostrophic flow: Comparison with the two-dimensional Euler flow, Nonlinear Phenomena in Ocean Dynamics (Los Alamos, NM, 1995), Phys. D., 98 (1996), 515-522. doi: 10.1016/0167-2789(96)00114-5.

[22]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.

[23]

D. Motreanu and V. Radulescu, Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems. Nonconvex Optimization and its Applications, 67. Kluwer Academic Publishers, Dordrecht, 2003. doi: 10.1007/978-1-4757-6921-0.

[24]

R. Musina and A. I. Nazarov, Variational inequalities for the spectral fractional Laplacian, Comp. Math. Math. Phy., 57 (2017), 373-386. doi: 10.1134/S0965542517030113.

[25]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[26]

O. Savin and E. Valdinoci, Elliptic PDEs with fibered nonlinearities, J. Geom. Anal., 19 (2009), 420-432. doi: 10.1007/s12220-008-9064-5.

[27]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[28]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Ibero., 29 (2013), 1091-1126. doi: 10.4171/RMI/750.

[29]

A. Signorini, Questioni di elasticita non linearizzata e semilinearizzata, Rendiconti di Matematica e Delle Sue Applicazioni, 18 (1959), 95-139.

[30]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.

[31]

J. Stoker, Water Waves: The Mathematical Theory with Applications, Pure and Applied Mathematics, Vol. IV., Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957.

[32]

J. Tan, Positive solutions for non local elliptic problems, Disc. Cont. Dyna. Sys., 33 (2013), 837-859. doi: 10.3934/dcds.2013.33.837.

[33]

K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Analysis: Real World Applications, 14 (2013), 867-874. doi: 10.1016/j.nonrwa.2012.08.008.

[34]

J. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150. doi: 10.1006/jfan.1996.3016.

[35]

L. XiY. Huang and Y. Zhou, The multiplicity of nontrivial solutions for hemivariational inequalities involving nonlocal elliptic operators, Nonlinear Analysis: Real World Applications, 21 (2015), 87-98. doi: 10.1016/j.nonrwa.2014.06.009.

[36]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/A and II/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

References:
[1]

G. Alberti and G. Bellettini, A nonlocal anisotropic model for phase transitions. I. The optimal profile problem, Math. Ann., 310 (1998), 527-560. doi: 10.1007/s002080050159.

[2]

X. Cabre and J. Sola-Morales, Layer solutions in a half-space for boundary reactions, Comm. Pure Appl. Math., 58 (2005), 1678-1732. doi: 10.1002/cpa.20093.

[3]

L. Caffarelli, Further regularity for the Signorini problem, Comm. Partial Differential Equations, 4 (1979), 1067-1075. doi: 10.1080/03605307908820119.

[4]

L. CaffarelliJ. M. Roquejoffre and Y. Sire, Variational problems in free boundaries for the fractional Laplacian, J. Eur. Math. Soc., 12 (2010), 1151-1179. doi: 10.4171/JEMS/226.

[5]

L. Caffarelli and A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, Annals of Mathematics. Second Series, 171 (2010), 1903-1930. doi: 10.4007/annals.2010.171.1903.

[6]

K. C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl., 80 (1981), 102-129. doi: 10.1016/0022-247X(81)90095-0.

[7]

F. H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

[8]

N. Costea and C. Varga, Systems of nonlinear hemivariational inequalities and applications, Topological Methods in Nonlinear Analysis, 41 (2013), 39-65.

[9]

W. Craig and M. Groves, Hamiltonian long-wave approximations to the water-wave problem, Wave Motion, 19 (1994), 367-389. doi: 10.1016/0165-2125(94)90003-5.

[10]

W. CraigU. Schanz and C. Sulem, The modulational regime of three-dimensional water waves and the Davey-Stewartson system, Ann. Inst. H. Poincare, Anal. Non Lineaire, 14 (1997), 615-667. doi: 10.1016/S0294-1449(97)80128-X.

[11]

L. Gasinski and N. S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Chapman and Hall/CRC, Boca Raton, London, New York, Washington, DC, 2005.

[12]

S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer, Dordrecht, The Netherlands, 1997. doi: 10.1007/978-1-4615-6359-4.

[13]

Z. H. Liu, A class of evolution hamivariational inequalities, Nonlinear Analysis, TMA, 36 (1999), 91-100. doi: 10.1016/S0362-546X(98)00016-9.

[14]

Z. H. Liu, Browder-Tikhonov regularization of non-coercive evolution hemivariational inequalities, Inverse Problems, 21 (2005), 13-20. doi: 10.1088/0266-5611/21/1/002.

[15]

Z. H. Liu, Existence results for quasilinear parabolic hemivariational inequalities, J. Differential Equations, 244 (2008), 1395-1409. doi: 10.1016/j.jde.2007.09.001.

[16]

Z. H. LiuX. W. Li and D. Motreanu, Approximate Controllability for Nonlinear Evolution Hemivariational Inequalities in Hilbert Spaces, SIAM Journal on Control and Optimization, 53 (2015), 3228-3244. doi: 10.1137/140994058.

[17]

Z. H. Liu and X. W. Li, Approximate controllability for a class of hemivariational inequalities, Nonlinear Analysis: Real World Applications, 22 (2015), 581-591. doi: 10.1016/j.nonrwa.2014.08.010.

[18]

Z. H. Liu, Anti-periodic solutions to nonlinear evolution equations, Journal of functional analysis, 258 (2010), 2026-2033. doi: 10.1016/j.jfa.2009.11.018.

[19]

Z. H. Liu and J. Tan, Nonlocal Elliptic Hemivariational inequalities, Electronic Journal of Qualitative Theory of Differential Equations, 66 (2017), 1-7. doi: 10.14232/ejqtde.2017.1.66.

[20]

Z. H. Liu and S. S. Zhang, On the degree theory for multivalued ($S_+$) type mappings, Applied Mathematics and Mechanics, 19 (1998), 1141-1149. doi: 10.1007/BF02456635.

[21]

A. Majda and E. Tabak, A two-dimensional model for quasigeostrophic flow: Comparison with the two-dimensional Euler flow, Nonlinear Phenomena in Ocean Dynamics (Los Alamos, NM, 1995), Phys. D., 98 (1996), 515-522. doi: 10.1016/0167-2789(96)00114-5.

[22]

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities. Models and Analysis of Contact Problems, Advances in Mechanics and Mathematics 26, Springer, New York, 2013. doi: 10.1007/978-1-4614-4232-5.

[23]

D. Motreanu and V. Radulescu, Variational and Non-Variational Methods in Nonlinear Analysis and Boundary Value Problems. Nonconvex Optimization and its Applications, 67. Kluwer Academic Publishers, Dordrecht, 2003. doi: 10.1007/978-1-4757-6921-0.

[24]

R. Musina and A. I. Nazarov, Variational inequalities for the spectral fractional Laplacian, Comp. Math. Math. Phy., 57 (2017), 373-386. doi: 10.1134/S0965542517030113.

[25]

E. Di NezzaG. Palatucci and E. Valdinoci, Hitchhikers guide to the fractional Sobolev spaces, Bull. Sci. Math., 136 (2012), 521-573. doi: 10.1016/j.bulsci.2011.12.004.

[26]

O. Savin and E. Valdinoci, Elliptic PDEs with fibered nonlinearities, J. Geom. Anal., 19 (2009), 420-432. doi: 10.1007/s12220-008-9064-5.

[27]

R. Servadei and E. Valdinoci, Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012), 887-898. doi: 10.1016/j.jmaa.2011.12.032.

[28]

R. Servadei and E. Valdinoci, Lewy-Stampacchia type estimates for variational inequalities driven by (non)local operators, Rev. Mat. Ibero., 29 (2013), 1091-1126. doi: 10.4171/RMI/750.

[29]

A. Signorini, Questioni di elasticita non linearizzata e semilinearizzata, Rendiconti di Matematica e Delle Sue Applicazioni, 18 (1959), 95-139.

[30]

Y. Sire and E. Valdinoci, Fractional Laplacian phase transitions and boundary reactions: A geometric inequality and a symmetry result, J. Funct. Anal., 256 (2009), 1842-1864. doi: 10.1016/j.jfa.2009.01.020.

[31]

J. Stoker, Water Waves: The Mathematical Theory with Applications, Pure and Applied Mathematics, Vol. IV., Interscience Publishers, Inc., New York; Interscience Publishers Ltd., London, 1957.

[32]

J. Tan, Positive solutions for non local elliptic problems, Disc. Cont. Dyna. Sys., 33 (2013), 837-859. doi: 10.3934/dcds.2013.33.837.

[33]

K. Teng, Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators, Nonlinear Analysis: Real World Applications, 14 (2013), 867-874. doi: 10.1016/j.nonrwa.2012.08.008.

[34]

J. Toland, The Peierls-Nabarro and Benjamin-Ono equations, J. Funct. Anal., 145 (1997), 136-150. doi: 10.1006/jfan.1996.3016.

[35]

L. XiY. Huang and Y. Zhou, The multiplicity of nontrivial solutions for hemivariational inequalities involving nonlocal elliptic operators, Nonlinear Analysis: Real World Applications, 21 (2015), 87-98. doi: 10.1016/j.nonrwa.2014.06.009.

[36]

E. Zeidler, Nonlinear Functional Analysis and Its Applications, II/A and II/B, Springer, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

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