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Variational analysis for nonlocal Yamabe-type systems
Blow up and blow up time for degenerate Kirchhoff-type wave problems involving the fractional Laplacian with arbitrary positive initial energy
1. | College of Automation, Harbin Engineering University, Heilongjiang, Harbin 150001, China |
2. | College of Mathematical Sciences, Harbin Engineering University, Heilongjiang, Harbin 150001, China |
$\left\{ \begin{align} & {{u}_{tt}}+[u]_{s}^{2(\theta -1)}{{(-\Delta )}^{s}}u=f(u),\ \ \ \ \text{in}\ \Omega \times {{\mathbb{R}}^{+}}, \\ & u(x,0)={{u}_{0}},\ \ {{u}_{t}}(x,0)={{u}_{1}},\ \ \ \ \ \ \text{in}\ \Omega , \\ & u=0,\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{in}\ ({{\mathbb{R}}^{N}}\backslash \Omega )\times \mathbb{R}_{0}^{+}, \\ \end{align} \right.$ |
$ [u]_s $ |
$ u $ |
$ s\in(0, 1) $ |
$ \theta\in[1, 2_s^*/2) $ |
$ 2_s^* = \frac{2N}{N-2s} $ |
$ (-\Delta)^s $ |
$ f(u) $ |
$ \Omega\subset\mathbb{R}^N $ |
$ \partial \Omega $ |
$ f(u) = |u|^{p-1}u $ |
References:
[1] |
D. Applebaum,
Lévy processes from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
|
[2] |
G. Autuori, P. Pucci and M. C. Salvatori,
Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.
doi: 10.1007/s00205-009-0241-x. |
[3] |
G. Autuori and P. Pucci,
Elliptic problems involving the fractional Laplacian in $\mathbb{R}^N$, J. Differential Equations, 255 (2013), 2340-2362.
doi: 10.1016/j.jde.2013.06.016. |
[4] |
G. M. Bisci and V. D. Rǎdulescu, Applications of local linking to nonlocal Neumann problems, Commun. Contemp. Math., 17 (2015), 1450001, 17 pp.
doi: 10.1142/S0219199714500011. |
[5] |
G. M. Bisci and V. D. Rǎdulescu,
Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.
doi: 10.1007/s00526-015-0891-5. |
[6] |
G. M. Bisci, V. D. Rǎdulescu 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.![]() ![]() ![]() |
[7] |
G. M. Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088, 23 pp.
doi: 10.1142/S0219199715500881. |
[8] |
M. Bonforte, Y. Sire and J. Vázquez,
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.
doi: 10.3934/dcds.2015.35.5725. |
[9] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[10] |
M. Caponi and P. Pucci,
Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.
doi: 10.1007/s10231-016-0555-x. |
[11] |
P. D'Ancona and S. Spagnolo,
Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.
doi: 10.1007/BF02100605. |
[12] |
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. |
[13] |
Y. Q. Fu and P. Pucci, On solutions of space-fractional diffusion equations by means of potential wells, Electron. J. Qual. Theory Differ. Equ., 70 (2016), 17 pp.
doi: 10.14232/ejqtde.2016.1.70. |
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F. Gazzola and T. Weth,
Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, 18 (2005), 961-990.
|
[15] |
F. Gazzola and M. Squassina,
Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.
doi: 10.1016/j.anihpc.2005.02.007. |
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M. Korpusov,
Non-existence of global solutions to generalized dissipative Klein-Gordon equations with positive energy, Electron. J. Differential Equations, 2012 (2012), 1-10.
|
[17] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[18] |
W. Lian, M. S. Ahmed and R. Z. Xu,
Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal., 184 (2019), 239-257.
doi: 10.1016/j.na.2019.02.015. |
[19] |
W. Lian, R. Z. Xu, V. D. 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). |
[20] |
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. |
[21] |
Y. C. Liu,
On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.
doi: 10.1016/S0022-0396(02)00020-7. |
[22] |
Y. C. Liu and R. Z. Xu,
Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.
doi: 10.3934/dcdsb.2007.7.171. |
[23] |
T. Matsuyama and R. Ikehata,
On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl., 204 (1996), 729-753.
doi: 10.1006/jmaa.1996.0464. |
[24] |
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. |
[25] |
N. Pan, P. Pucci and B. L. Zhang,
Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian, J. Evol. Equ., 18 (2018), 385-409.
doi: 10.1007/s00028-017-0406-2. |
[26] |
L. E. Pany and D. H. Sattinger,
Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.
doi: 10.1007/BF02761595. |
[27] |
N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Cham, 2019.
doi: 10.1007/978-3-030-03430-6. |
[28] |
P. Pucci and V. D. Rǎdulescu,
Progress in nonlinear Kirchhoff problems, Nonlinear Anal., 186 (2019), 1-5.
doi: 10.1016/j.na.2019.02.022. |
[29] |
P. Pucci and S. Saldi,
Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.
doi: 10.4171/RMI/879. |
[30] |
P. Pucci, M. 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. |
[31] |
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. |
[32] |
J. L. Vázquez,
Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.
doi: 10.3934/dcdss.2014.7.857. |
[33] |
J. L. Vázquez,
Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D, J. Evol. Equ., 16 (2016), 723-758.
doi: 10.1007/s00028-016-0340-8. |
[34] |
M. Q. Xiang, B. L. Zhang and V. D. Rǎdulescu,
Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 3186-3205.
doi: 10.1088/0951-7715/29/10/3186. |
[35] |
R. Z. Xu,
Initial boundary value problem of semilinear hyperbolic equations and parabolic equations with critical initial data, Quar. Appl. Math., 68 (2010), 459-468.
doi: 10.1090/S0033-569X-2010-01197-0. |
[36] |
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. |
[37] |
R. Z. Xu, M. Y. Zhang, S. H. Chen, Y. B. Yang and J. H. Shen,
The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.
doi: 10.3934/dcds.2017244. |
[38] |
R. Z. Xu, Y. X. Chen, Y. B. Yang, S. H. Chen, J. H. Shen, T. Yu and Z. S. Xu, Global well-posedness of semilinear hyperbolic equations, parabolic equations and schrodinger equations, Electron. J. Differential Equations, 55 (2018), 52 pp.
doi: 10.3934/dcds.2017244. |
[39] |
R. Z. Xu, X. C. Wang and Y. B. Yang,
Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176-181.
doi: 10.1016/j.aml.2018.03.033. |
[40] |
Y. B. Yang and R. Z. Xu,
Finite time blowup for nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Appl. Math. Lett., 77 (2018), 21-26.
doi: 10.1016/j.aml.2017.09.014. |
[41] |
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. |
[42] |
R. Z. Xu, W. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., (2019), https://doi.org/10.1007/s11425-017-9280-x.
doi: 10.1007/s11425-017-9280-x. |
[43] |
T. Yamazaki,
Scattering for a quasilinear hyperbolic equation of Kirchhoff type, J. Differential Equations, 143 (1998), 1-59.
doi: 10.1006/jdeq.1997.3372. |
[44] |
B. L. Zhang, V. D. Rǎdulescu and L. Wang,
Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 1061-1081.
doi: 10.1080/17476933.2015.1005612. |
show all references
References:
[1] |
D. Applebaum,
Lévy processes from probability to finance quantum groups, Notices Amer. Math. Soc., 51 (2004), 1336-1347.
|
[2] |
G. Autuori, P. Pucci and M. C. Salvatori,
Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.
doi: 10.1007/s00205-009-0241-x. |
[3] |
G. Autuori and P. Pucci,
Elliptic problems involving the fractional Laplacian in $\mathbb{R}^N$, J. Differential Equations, 255 (2013), 2340-2362.
doi: 10.1016/j.jde.2013.06.016. |
[4] |
G. M. Bisci and V. D. Rǎdulescu, Applications of local linking to nonlocal Neumann problems, Commun. Contemp. Math., 17 (2015), 1450001, 17 pp.
doi: 10.1142/S0219199714500011. |
[5] |
G. M. Bisci and V. D. Rǎdulescu,
Ground state solutions of scalar field fractional Schrödinger equations, Calc. Var. Partial Differential Equations, 54 (2015), 2985-3008.
doi: 10.1007/s00526-015-0891-5. |
[6] |
G. M. Bisci, V. D. Rǎdulescu 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.![]() ![]() ![]() |
[7] |
G. M. Bisci and L. Vilasi, On a fractional degenerate Kirchhoff-type problem, Commun. Contemp. Math., 19 (2017), 1550088, 23 pp.
doi: 10.1142/S0219199715500881. |
[8] |
M. Bonforte, Y. Sire and J. Vázquez,
Existence, uniqueness and asymptotic behaviour for fractional porous medium equations on bounded domains, Discrete Contin. Dyn. Syst., 35 (2015), 5725-5767.
doi: 10.3934/dcds.2015.35.5725. |
[9] |
L. Caffarelli and L. Silvestre,
An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007), 1245-1260.
doi: 10.1080/03605300600987306. |
[10] |
M. Caponi and P. Pucci,
Existence theorems for entire solutions of stationary Kirchhoff fractional $p$-Laplacian equations, Ann. Mat. Pura Appl., 195 (2016), 2099-2129.
doi: 10.1007/s10231-016-0555-x. |
[11] |
P. D'Ancona and S. Spagnolo,
Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. Math., 108 (1992), 247-262.
doi: 10.1007/BF02100605. |
[12] |
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. |
[13] |
Y. Q. Fu and P. Pucci, On solutions of space-fractional diffusion equations by means of potential wells, Electron. J. Qual. Theory Differ. Equ., 70 (2016), 17 pp.
doi: 10.14232/ejqtde.2016.1.70. |
[14] |
F. Gazzola and T. Weth,
Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level, Differential Integral Equations, 18 (2005), 961-990.
|
[15] |
F. Gazzola and M. Squassina,
Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.
doi: 10.1016/j.anihpc.2005.02.007. |
[16] |
M. Korpusov,
Non-existence of global solutions to generalized dissipative Klein-Gordon equations with positive energy, Electron. J. Differential Equations, 2012 (2012), 1-10.
|
[17] |
N. Laskin, Fractional Schrödinger equation, Phys. Rev. E, 66 (2002), 056108, 7 pp.
doi: 10.1103/PhysRevE.66.056108. |
[18] |
W. Lian, M. S. Ahmed and R. Z. Xu,
Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity, Nonlinear Anal., 184 (2019), 239-257.
doi: 10.1016/j.na.2019.02.015. |
[19] |
W. Lian, R. Z. Xu, V. D. 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). |
[20] |
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. |
[21] |
Y. C. Liu,
On potential wells and vacuum isolating of solutions for semilinear wave equations, J. Differential Equations, 192 (2003), 155-169.
doi: 10.1016/S0022-0396(02)00020-7. |
[22] |
Y. C. Liu and R. Z. Xu,
Wave equations and reaction-diffusion equations with several nonlinear source terms of different sign, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 171-189.
doi: 10.3934/dcdsb.2007.7.171. |
[23] |
T. Matsuyama and R. Ikehata,
On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms, J. Math. Anal. Appl., 204 (1996), 729-753.
doi: 10.1006/jmaa.1996.0464. |
[24] |
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. |
[25] |
N. Pan, P. Pucci and B. L. Zhang,
Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian, J. Evol. Equ., 18 (2018), 385-409.
doi: 10.1007/s00028-017-0406-2. |
[26] |
L. E. Pany and D. H. Sattinger,
Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.
doi: 10.1007/BF02761595. |
[27] |
N. S. Papageorgiou, V. D. Rǎdulescu and D. D. Repovš, Nonlinear Analysis-Theory and Methods, Springer Monographs in Mathematics. Springer, Cham, 2019.
doi: 10.1007/978-3-030-03430-6. |
[28] |
P. Pucci and V. D. Rǎdulescu,
Progress in nonlinear Kirchhoff problems, Nonlinear Anal., 186 (2019), 1-5.
doi: 10.1016/j.na.2019.02.022. |
[29] |
P. Pucci and S. Saldi,
Critical stationary Kirchhoff equations in $\mathbb{R}^N$ involving nonlocal operators, Rev. Mat. Iberoam., 32 (2016), 1-22.
doi: 10.4171/RMI/879. |
[30] |
P. Pucci, M. 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. |
[31] |
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. |
[32] |
J. L. Vázquez,
Recent progress in the theory of nonlinear diffusion with fractional Laplacian operators, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), 857-885.
doi: 10.3934/dcdss.2014.7.857. |
[33] |
J. L. Vázquez,
Existence of maximal solutions for some very singular nonlinear fractional diffusion equations in 1D, J. Evol. Equ., 16 (2016), 723-758.
doi: 10.1007/s00028-016-0340-8. |
[34] |
M. Q. Xiang, B. L. Zhang and V. D. Rǎdulescu,
Multiplicity of solutions for a class of quasilinear Kirchhoff system involving the fractional $p$-Laplacian, Nonlinearity, 29 (2016), 3186-3205.
doi: 10.1088/0951-7715/29/10/3186. |
[35] |
R. Z. Xu,
Initial boundary value problem of semilinear hyperbolic equations and parabolic equations with critical initial data, Quar. Appl. Math., 68 (2010), 459-468.
doi: 10.1090/S0033-569X-2010-01197-0. |
[36] |
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. |
[37] |
R. Z. Xu, M. Y. Zhang, S. H. Chen, Y. B. Yang and J. H. Shen,
The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), 5631-5649.
doi: 10.3934/dcds.2017244. |
[38] |
R. Z. Xu, Y. X. Chen, Y. B. Yang, S. H. Chen, J. H. Shen, T. Yu and Z. S. Xu, Global well-posedness of semilinear hyperbolic equations, parabolic equations and schrodinger equations, Electron. J. Differential Equations, 55 (2018), 52 pp.
doi: 10.3934/dcds.2017244. |
[39] |
R. Z. Xu, X. C. Wang and Y. B. Yang,
Blowup and blowup time for a class of semilinear pseudo-parabolic equations with high initial energy, Appl. Math. Lett., 83 (2018), 176-181.
doi: 10.1016/j.aml.2018.03.033. |
[40] |
Y. B. Yang and R. Z. Xu,
Finite time blowup for nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Appl. Math. Lett., 77 (2018), 21-26.
doi: 10.1016/j.aml.2017.09.014. |
[41] |
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. |
[42] |
R. Z. Xu, W. Lian and Y. Niu, Global well-posedness of coupled parabolic systems, Sci. China Math., (2019), https://doi.org/10.1007/s11425-017-9280-x.
doi: 10.1007/s11425-017-9280-x. |
[43] |
T. Yamazaki,
Scattering for a quasilinear hyperbolic equation of Kirchhoff type, J. Differential Equations, 143 (1998), 1-59.
doi: 10.1006/jdeq.1997.3372. |
[44] |
B. L. Zhang, V. D. Rǎdulescu and L. Wang,
Existence results for Kirchhoff-type superlinear problems involving the fractional Laplacian, Proc. Roy. Soc. Edinburgh Sect. A, 149 (2019), 1061-1081.
doi: 10.1080/17476933.2015.1005612. |
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