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June  2020, 28(2): 651-669. doi: 10.3934/era.2020034

Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping

1. 

College of Science, Civil Aviation University of China, Tianjin 300300, China

2. 

College of Mathematics and System Science, Shandong University of Science and Technology, Qingdao 266590, China

* Corresponding author: Binlin Zhang

Received  December 2019 Revised  March 2020 Published  April 2020

The aim of this paper is to investigate the existence of weak solutions for a Kirchhoff-type differential inclusion wave problem involving a discontinuous set-valued term, the fractional $ p $-Laplacian and linear strong damping term. The existence of weak solutions is obtained by using a regularization method combined with the Galerkin method.

Citation: Mingqi Xiang, Binlin Zhang, Die Hu. Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping. Electronic Research Archive, 2020, 28 (2) : 651-669. doi: 10.3934/era.2020034
References:
[1]

S. Antontsev, Wave equation with $p(x, t)$-Laplacian and damping term: Existence and blow-up, Differ. Equations Appl., 3 (2011), 503-525.  doi: 10.7153/dea-03-32.  Google Scholar

[2]

F. V. Atkinson and F. A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations, 70 (1987), 349-365.  doi: 10.1016/0022-0396(87)90156-2.  Google Scholar

[3]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Rational Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar

[4]

L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, Springer, Heidelberg, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

[5]

S. T. ChenB. L. Zhang and X. H. Tang, Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity, Adv. Nonlinear Anal., 9 (2020), 148-167.  doi: 10.1515/anona-2018-0147.  Google Scholar

[6]

J. Clements, Existence theorems for a quasilinear evolution equation, SIAM J. Appl. Math., 26 (1974), 745-752.  doi: 10.1137/0126066.  Google Scholar

[7]

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.  Google Scholar

[8]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

[9]

A. Friedman and J. Nečas, Systems of nonlinear wave equations with nonlinear viscosity, Pac. J. Math., 135 (1988), 29-55.  doi: 10.2140/pjm.1988.135.29.  Google Scholar

[10]

Y. Q. Fu and N. Pan, Existence of solutions for nonlinear parabolic problems with $p(x)$-growth, J. Math. Anal. Appl., 362 (2010), 313-326.  doi: 10.1016/j.jmaa.2009.08.038.  Google Scholar

[11]

C. JiF. Fang and B. L. Zhang, A multiplicity result for asymptotically linear Kirchhoff equations, Adv. Nonlinear Anal., 8 (2019), 267-277.  doi: 10.1515/anona-2016-0240.  Google Scholar

[12]

G. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[13]

O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23. American Mathematical Society, Providence, R.I., 1968.  Google Scholar

[14]

R. Landes, On the existence of weak solutions for quasilinear parabolic initial boundary value problem, Proc. Roy. Soc. Edinburgh Sect. A, 89 (1981), 217-237.  doi: 10.1017/S0308210500020242.  Google Scholar

[15]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[16]

W. Lian and R. Z. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[17]

W. Lian, R. Z. Xu, V. Rǎdulescu, Y. B. Yang and N. Zhao, Global well-posedness for a class of fourth order nonlinear strongly damped wave equations, Adv. Calc. Var., (2019). doi: 10.1515/acv-2019-0039.  Google Scholar

[18]

S. H. LiangD. D. Repovš and B. L. Zhang, Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities, Math. Method. Appl. Sci., 43 (2020), 2473-2490.  doi: 10.1002/mma.6057.  Google Scholar

[19]

S. H. LiangD. Repovš and B. L. Zhang, On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity, Comput. Math. Appl., 75 (2018), 1778-1794.  doi: 10.1016/j.camwa.2017.11.033.  Google Scholar

[20]

Q. Lin, X. T. Tian, R. Z. Xu and M. N. Zhang, Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy, Discrete Contin. Dyn. Syst. Ser. S, (2019). doi: 10.3934/dcdss.2020160.  Google Scholar

[21]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[22]

J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. Fr., 93 (1965), 43-96.   Google Scholar

[23]

T. F. Ma and J. A. Soriano, On weak solutions for an evolution equation with exponent nonlinearities, Nonlinear Anal., 37 (1977), 1029-1038.  doi: 10.1016/S0362-546X(97)00714-1.  Google Scholar

[24]

M. Q. XiangV. D. Rǎdulescu and B. L. Zhang, Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250.  doi: 10.1088/1361-6544/aaba35.  Google Scholar

[25]

G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397.  Google Scholar

[26]

M. Q. Xiang, V. D. Rǎdulescu and B. L. Zhang, A critical fractional Choquard-Kirchhoff problem with magnetic field, Comm. Contem. Math., 21 (2019), 1850004, 36 pp. doi: 10.1142/s0219199718500049.  Google Scholar

[27]

M. Q. Xiang, V. Rǎdulescu and B. L. Zhang, Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity, Cal. Var. Partial Differential Equations, 58 (2019), 27 pp. doi: 10.1007/s00526-019-1499-y.  Google Scholar

[28]

N. PanP. PucciR. Z. Xu and B. L. Zhang, Degenerate Kirchhoff-type wave problems involving the fractional Laplacian with nonlinear damping and source terms, J. Evolution Equations, 19 (2019), 615-643.  doi: 10.1007/s00028-019-00489-6.  Google Scholar

[29]

N. PanP. Pucci and B. L. Zhang, Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian, J. Evolution Equations, 18 (2018), 385-409.  doi: 10.1007/s00028-017-0406-2.  Google Scholar

[30]

N. PanB. L. Zhang and J. Cao, Degenerate Kirchhoff-type diffusion problems involving the fractional $p$-Laplacian, Nonlinear Anal. Real World Appl., 37 (2017), 56-70.  doi: 10.1016/j.nonrwa.2017.02.004.  Google Scholar

[31]

N. S. Papageorgiou, V. Rǎdulescu and D. Repovš, Relaxation methods for optimal control problems, Bull. Math. Sci., (2020). doi: 10.1142/S1664360720500046.  Google Scholar

[32]

J. Y. ParkH. M. Kim and S. H. Park, On weak solutions for hyperbolic differential inclusion with discontinuous nonlinearities, Nonlinear Anal., 55 (2003), 103-113.  doi: 10.1016/S0362-546X(03)00216-5.  Google Scholar

[33]

P. PucciM. Q. Xiang and B. L. Zhang, A diffusion problem of Kirchhoff type involving the nonlocal fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 37 (2017), 4035-4051.  doi: 10.3934/dcds.2017171.  Google Scholar

[34]

P. PucciM. Q. Xiang and B. L. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102.  Google Scholar

[35]

G. Singh, Nonlocal perturbations of the fractional Choquard equation, Adv. Nonlinear Anal., 8 (2019), 694-706.  doi: 10.1515/anona-2017-0126.  Google Scholar

[36]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symp., Springer, Heidelberg, 7 (2012), 271–298. doi: 10.1007/978-3-642-25361-4_15.  Google Scholar

[37]

J. Clments, Existence theorems for a quasilinear evolution equation, SIAM J. Appl. Math., 26 (1974), 745-752.  doi: 10.1137/0126066.  Google Scholar

[38]

M. Q. XiangB. L. Zhang and D. Yang, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal., 178 (2019), 190-204.  doi: 10.1016/j.na.2018.07.016.  Google Scholar

[39]

R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[40]

R. Z. Xu, X. C. Wang, Y. B. Yang and S. H. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.  Google Scholar

[41]

R. Z. XuM. Y. ZhangS. H. ChenY. B. Yang and J. H. Shen, The initial-boundary value problems for a class of six order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

show all references

References:
[1]

S. Antontsev, Wave equation with $p(x, t)$-Laplacian and damping term: Existence and blow-up, Differ. Equations Appl., 3 (2011), 503-525.  doi: 10.7153/dea-03-32.  Google Scholar

[2]

F. V. Atkinson and F. A. Peletier, Elliptic equations with nearly critical growth, J. Differential Equations, 70 (1987), 349-365.  doi: 10.1016/0022-0396(87)90156-2.  Google Scholar

[3]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Rational Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar

[4]

L. Caffarelli, Non-local diffusions, drifts and games, Nonlinear Partial Differential Equations, Abel Symposia, Springer, Heidelberg, 7 (2012), 37-52.  doi: 10.1007/978-3-642-25361-4_3.  Google Scholar

[5]

S. T. ChenB. L. Zhang and X. H. Tang, Existence and non-existence results for Kirchhoff-type problems with convolution nonlinearity, Adv. Nonlinear Anal., 9 (2020), 148-167.  doi: 10.1515/anona-2018-0147.  Google Scholar

[6]

J. Clements, Existence theorems for a quasilinear evolution equation, SIAM J. Appl. Math., 26 (1974), 745-752.  doi: 10.1137/0126066.  Google Scholar

[7]

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.  Google Scholar

[8]

A. Fiscella and E. Valdinoci, A critical Kirchhoff type problem involving a nonlocal operator, Nonlinear Anal., 94 (2014), 156-170.  doi: 10.1016/j.na.2013.08.011.  Google Scholar

[9]

A. Friedman and J. Nečas, Systems of nonlinear wave equations with nonlinear viscosity, Pac. J. Math., 135 (1988), 29-55.  doi: 10.2140/pjm.1988.135.29.  Google Scholar

[10]

Y. Q. Fu and N. Pan, Existence of solutions for nonlinear parabolic problems with $p(x)$-growth, J. Math. Anal. Appl., 362 (2010), 313-326.  doi: 10.1016/j.jmaa.2009.08.038.  Google Scholar

[11]

C. JiF. Fang and B. L. Zhang, A multiplicity result for asymptotically linear Kirchhoff equations, Adv. Nonlinear Anal., 8 (2019), 267-277.  doi: 10.1515/anona-2016-0240.  Google Scholar

[12]

G. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanik, Teubner, Leipzig, 1883. Google Scholar

[13]

O. A. Ladyžhenskaja, V. A. Solonnikov and N. N. Ural'tseva, Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, 23. American Mathematical Society, Providence, R.I., 1968.  Google Scholar

[14]

R. Landes, On the existence of weak solutions for quasilinear parabolic initial boundary value problem, Proc. Roy. Soc. Edinburgh Sect. A, 89 (1981), 217-237.  doi: 10.1017/S0308210500020242.  Google Scholar

[15]

N. Laskin, Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.  doi: 10.1016/S0375-9601(00)00201-2.  Google Scholar

[16]

W. Lian and R. Z. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), 613-632.  doi: 10.1515/anona-2020-0016.  Google Scholar

[17]

W. Lian, R. Z. Xu, V. Rǎdulescu, Y. B. Yang and N. Zhao, Global well-posedness for a class of fourth order nonlinear strongly damped wave equations, Adv. Calc. Var., (2019). doi: 10.1515/acv-2019-0039.  Google Scholar

[18]

S. H. LiangD. D. Repovš and B. L. Zhang, Fractional magnetic Schrödinger-Kirchhoff problems with convolution and critical nonlinearities, Math. Method. Appl. Sci., 43 (2020), 2473-2490.  doi: 10.1002/mma.6057.  Google Scholar

[19]

S. H. LiangD. Repovš and B. L. Zhang, On the fractional Schrödinger-Kirchhoff equations with electromagnetic fields and critical nonlinearity, Comput. Math. Appl., 75 (2018), 1778-1794.  doi: 10.1016/j.camwa.2017.11.033.  Google Scholar

[20]

Q. Lin, X. T. Tian, R. Z. Xu and M. N. Zhang, Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy, Discrete Contin. Dyn. Syst. Ser. S, (2019). doi: 10.3934/dcdss.2020160.  Google Scholar

[21]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Nonlinéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[22]

J.-L. Lions and W. A. Strauss, Some non-linear evolution equations, Bull. Soc. Math. Fr., 93 (1965), 43-96.   Google Scholar

[23]

T. F. Ma and J. A. Soriano, On weak solutions for an evolution equation with exponent nonlinearities, Nonlinear Anal., 37 (1977), 1029-1038.  doi: 10.1016/S0362-546X(97)00714-1.  Google Scholar

[24]

M. Q. XiangV. D. Rǎdulescu and B. L. Zhang, Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250.  doi: 10.1088/1361-6544/aaba35.  Google Scholar

[25]

G. Molica Bisci, V. D. Radulescu and R. Servadei, Variational Methods for Nonlocal Fractional Problems, Encyclopedia of Mathematics and its Applications, 162. Cambridge University Press, Cambridge, 2016. doi: 10.1017/CBO9781316282397.  Google Scholar

[26]

M. Q. Xiang, V. D. Rǎdulescu and B. L. Zhang, A critical fractional Choquard-Kirchhoff problem with magnetic field, Comm. Contem. Math., 21 (2019), 1850004, 36 pp. doi: 10.1142/s0219199718500049.  Google Scholar

[27]

M. Q. Xiang, V. Rǎdulescu and B. L. Zhang, Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity, Cal. Var. Partial Differential Equations, 58 (2019), 27 pp. doi: 10.1007/s00526-019-1499-y.  Google Scholar

[28]

N. PanP. PucciR. Z. Xu and B. L. Zhang, Degenerate Kirchhoff-type wave problems involving the fractional Laplacian with nonlinear damping and source terms, J. Evolution Equations, 19 (2019), 615-643.  doi: 10.1007/s00028-019-00489-6.  Google Scholar

[29]

N. PanP. Pucci and B. L. Zhang, Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian, J. Evolution Equations, 18 (2018), 385-409.  doi: 10.1007/s00028-017-0406-2.  Google Scholar

[30]

N. PanB. L. Zhang and J. Cao, Degenerate Kirchhoff-type diffusion problems involving the fractional $p$-Laplacian, Nonlinear Anal. Real World Appl., 37 (2017), 56-70.  doi: 10.1016/j.nonrwa.2017.02.004.  Google Scholar

[31]

N. S. Papageorgiou, V. Rǎdulescu and D. Repovš, Relaxation methods for optimal control problems, Bull. Math. Sci., (2020). doi: 10.1142/S1664360720500046.  Google Scholar

[32]

J. Y. ParkH. M. Kim and S. H. Park, On weak solutions for hyperbolic differential inclusion with discontinuous nonlinearities, Nonlinear Anal., 55 (2003), 103-113.  doi: 10.1016/S0362-546X(03)00216-5.  Google Scholar

[33]

P. PucciM. Q. Xiang and B. L. Zhang, A diffusion problem of Kirchhoff type involving the nonlocal fractional $p$-Laplacian, Discrete Contin. Dyn. Syst., 37 (2017), 4035-4051.  doi: 10.3934/dcds.2017171.  Google Scholar

[34]

P. PucciM. Q. Xiang and B. L. Zhang, Existence and multiplicity of entire solutions for fractional $p$-Kirchhoff equations, Adv. Nonlinear Anal., 5 (2016), 27-55.  doi: 10.1515/anona-2015-0102.  Google Scholar

[35]

G. Singh, Nonlocal perturbations of the fractional Choquard equation, Adv. Nonlinear Anal., 8 (2019), 694-706.  doi: 10.1515/anona-2017-0126.  Google Scholar

[36]

J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, Nonlinear Partial Differential Equations, Abel Symp., Springer, Heidelberg, 7 (2012), 271–298. doi: 10.1007/978-3-642-25361-4_15.  Google Scholar

[37]

J. Clments, Existence theorems for a quasilinear evolution equation, SIAM J. Appl. Math., 26 (1974), 745-752.  doi: 10.1137/0126066.  Google Scholar

[38]

M. Q. XiangB. L. Zhang and D. Yang, Multiplicity results for variable-order fractional Laplacian equations with variable growth, Nonlinear Anal., 178 (2019), 190-204.  doi: 10.1016/j.na.2018.07.016.  Google Scholar

[39]

R. Z. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.  doi: 10.1016/j.jfa.2013.03.010.  Google Scholar

[40]

R. Z. Xu, X. C. Wang, Y. B. Yang and S. H. Chen, Global solutions and finite time blow-up for fourth order nonlinear damped wave equation, J. Math. Phys., 59 (2018), 061503, 27 pp. doi: 10.1063/1.5006728.  Google Scholar

[41]

R. Z. XuM. Y. ZhangS. H. ChenY. B. Yang and J. H. Shen, The initial-boundary value problems for a class of six order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.  doi: 10.3934/dcds.2017244.  Google Scholar

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