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Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources
Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity
Department of Mathematics, College of Science, Civil Aviation University of China, Tianjin, 300300, China |
$ \begin{align*} u_{tt}+M([u]_{\alpha, 2}^2)(-\Delta)^{\alpha}u+(-\Delta)^{s}u_{t} = \int_{0}^{t}g(t-\tau)(-\Delta)^{\alpha}u(\tau)d\tau+\lambda|u|^{q -2}u, \end{align*} $ |
$ M:[0, \infty)\rightarrow (0, \infty) $ |
$ [u]_{\alpha, 2} $ |
$ u $ |
$ (-\Delta)^\alpha $ |
$ (-\Delta)^s $ |
$ g:\mathbb{R}^+\rightarrow \mathbb{R}^+ $ |
$ \lambda $ |
References:
[1] |
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. |
[2] |
L. Caffarelli, Non-local diffusions, drifts and games, in Nonlinear Partial Differential Equations, Abel Symp., 7, Springer, Heidelberg, 2012, 37–52.
doi: 10.1007/978-3-642-25361-4_3. |
[3] |
M. Can, S. R. Park and F. Aliyev,
Nonexistence of global solutions of some quasilinear hyperbolic equations, J. Math. Anal. Appl., 213 (1997), 540-553.
doi: 10.1006/jmaa.1997.5557. |
[4] |
A. Castro and S.-Z. Song,
Infinitely many radial solutions for a super-cubic Kirchhoff type problem in a ball, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3347-3355.
doi: 10.3934/dcdss.2020127. |
[5] |
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. |
[6] |
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. |
[7] |
A. Friedman and J. Neǎas,
Systems of nonlinear wave equations with nonlinear viscosity, Pacific J. Math., 135 (1988), 29-55.
doi: 10.2140/pjm.1988.135.29. |
[8] |
Y. 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. |
[9] |
X. Han and M. Wang,
Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Anal., 71 (2009), 5427-5450.
doi: 10.1016/j.na.2009.04.031. |
[10] |
P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, Mass., 1982. |
[11] |
J. A. Kim and Y. H. Han,
Blow up of solution of a nonlinear viscoelastic wave equation, Acta Appl. Math., 111 (2010), 1-6.
doi: 10.1007/s10440-009-9524-3. |
[12] |
G. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanik, Teubner, Leipzig, 1883. |
[13] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[14] |
H. A. Levine,
Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_tt = Au+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.
doi: 10.2307/1996814. |
[15] |
W. Lian, V. D. Rǎdulescu, R. Xu, Y. Yang and N. Zhao,
Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Adv. Calc. Var., 14 (2021), 589-611.
doi: 10.1515/acv-2019-0039. |
[16] |
W. Lian, J. Wang and R. Xu,
Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.
doi: 10.1016/j.jde.2020.03.047. |
[17] |
W. Lian and R. 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. |
[18] |
Q. Lin, X. Tian, R. Xu and M. 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, 13 (2020), 2095-2107.
doi: 10.3934/dcdss.2020160. |
[19] |
J.-L. Lions and W. A. Strauss,
Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96.
doi: 10.24033/bsmf.1616. |
[20] |
G. Liu,
The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.
doi: 10.3934/era.2020016. |
[21] |
Y. Liu,
Long-time behavior of a class of viscoelastic plate equations, Electron. Res. Arch., 28 (2020), 311-326.
doi: 10.3934/era.2020018. |
[22] |
S. A. Messaoudi,
Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320 (2006), 902-915.
doi: 10.1016/j.jmaa.2005.07.022. |
[23] |
N. Pan, P. Pucci and B. Zhang,
Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian, J. Evol. Equ., 18 (2018), 385-409.
doi: 10.1007/s00028-017-0406-2. |
[24] |
P. Pucci, M. Xiang and B. 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. |
[25] |
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997.
doi: 10.1090/surv/049. |
[26] |
H. Song and D. Xue,
Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Anal., 109 (2014), 245-251.
doi: 10.1016/j.na.2014.06.012. |
[27] |
J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in Nonlinear Partial Differential Equations, Abel Symp., 7, Springer, Heidelberg, 2012,271–298.
doi: 10.1007/978-3-642-25361-4_15. |
[28] |
F. Wang, D. Hu and M. Xiang,
Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems, Adv. Nonlinear Anal., 10 (2021), 636-658.
doi: 10.1515/anona-2020-0150. |
[29] |
X. Wang and R. Xu,
Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.
doi: 10.1515/anona-2020-0141. |
[30] |
M. Xiang, G. M. Bisci and B. Zhang,
Variational analysis for nonlocal Yamabe-type systems, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2069-2094.
doi: 10.3934/dcdss.2020159. |
[31] |
M. Xiang, D. Hu, B. Zhang and Y. Wang, Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growth, J. Math. Anal. Appl., 501 (2021), 19pp.
doi: 10.1016/j.jmaa.2020.124269. |
[32] |
M. Xiang, V. D. Rǎdulescu and B. Zhang, A critical fractional Choquard-Kirchhoff problem with magnetic field, Commun. Contemp. Math., 21 (2019), 36pp.
doi: 10.1142/s0219199718500049. |
[33] |
M. Xiang, V. D. Rǎdulescu and B. Zhang, Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity, Cal. Var. Partial Differential Equations, 58 (2019), 27pp.
doi: 10.1007/s00526-019-1499-y. |
[34] |
M. Xiang, V. D. Rǎdulescu and B. Zhang,
Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250.
doi: 10.1088/1361-6544/aaba35. |
[35] |
M. Xiang, B. Zhang and M. Ferrara,
Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041.
doi: 10.1016/j.jmaa.2014.11.055. |
[36] |
M. Xiang, B. Zhang and D. Hu,
Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping, Electron. Res. Arch., 28 (2020), 651-669.
doi: 10.3934/era.2020034. |
[37] |
R. Xu, Y. Yang and Y. Liu,
Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.
doi: 10.1080/00036811.2011.601456. |
show all references
References:
[1] |
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. |
[2] |
L. Caffarelli, Non-local diffusions, drifts and games, in Nonlinear Partial Differential Equations, Abel Symp., 7, Springer, Heidelberg, 2012, 37–52.
doi: 10.1007/978-3-642-25361-4_3. |
[3] |
M. Can, S. R. Park and F. Aliyev,
Nonexistence of global solutions of some quasilinear hyperbolic equations, J. Math. Anal. Appl., 213 (1997), 540-553.
doi: 10.1006/jmaa.1997.5557. |
[4] |
A. Castro and S.-Z. Song,
Infinitely many radial solutions for a super-cubic Kirchhoff type problem in a ball, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 3347-3355.
doi: 10.3934/dcdss.2020127. |
[5] |
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. |
[6] |
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. |
[7] |
A. Friedman and J. Neǎas,
Systems of nonlinear wave equations with nonlinear viscosity, Pacific J. Math., 135 (1988), 29-55.
doi: 10.2140/pjm.1988.135.29. |
[8] |
Y. 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. |
[9] |
X. Han and M. Wang,
Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Anal., 71 (2009), 5427-5450.
doi: 10.1016/j.na.2009.04.031. |
[10] |
P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, Mass., 1982. |
[11] |
J. A. Kim and Y. H. Han,
Blow up of solution of a nonlinear viscoelastic wave equation, Acta Appl. Math., 111 (2010), 1-6.
doi: 10.1007/s10440-009-9524-3. |
[12] |
G. Kirchhoff, Vorlesungen über Mathematische Physik, Mechanik, Teubner, Leipzig, 1883. |
[13] |
N. Laskin,
Fractional quantum mechanics and Lévy path integrals, Phys. Lett. A, 268 (2000), 298-305.
doi: 10.1016/S0375-9601(00)00201-2. |
[14] |
H. A. Levine,
Instability and nonexistence of global solutions of nonlinear wave equation of the form $Pu_tt = Au+F(u)$, Trans. Amer. Math. Soc., 192 (1974), 1-21.
doi: 10.2307/1996814. |
[15] |
W. Lian, V. D. Rǎdulescu, R. Xu, Y. Yang and N. Zhao,
Global well-posedness for a class of fourth-order nonlinear strongly damped wave equations, Adv. Calc. Var., 14 (2021), 589-611.
doi: 10.1515/acv-2019-0039. |
[16] |
W. Lian, J. Wang and R. Xu,
Global existence and blow up of solutions for pseudo-parabolic equation with singular potential, J. Differential Equations, 269 (2020), 4914-4959.
doi: 10.1016/j.jde.2020.03.047. |
[17] |
W. Lian and R. 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. |
[18] |
Q. Lin, X. Tian, R. Xu and M. 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, 13 (2020), 2095-2107.
doi: 10.3934/dcdss.2020160. |
[19] |
J.-L. Lions and W. A. Strauss,
Some non-linear evolution equations, Bull. Soc. Math. France, 93 (1965), 43-96.
doi: 10.24033/bsmf.1616. |
[20] |
G. Liu,
The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term, Electron. Res. Arch., 28 (2020), 263-289.
doi: 10.3934/era.2020016. |
[21] |
Y. Liu,
Long-time behavior of a class of viscoelastic plate equations, Electron. Res. Arch., 28 (2020), 311-326.
doi: 10.3934/era.2020018. |
[22] |
S. A. Messaoudi,
Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation, J. Math. Anal. Appl., 320 (2006), 902-915.
doi: 10.1016/j.jmaa.2005.07.022. |
[23] |
N. Pan, P. Pucci and B. Zhang,
Degenerate Kirchhoff-type hyperbolic problems involving the fractional Laplacian, J. Evol. Equ., 18 (2018), 385-409.
doi: 10.1007/s00028-017-0406-2. |
[24] |
P. Pucci, M. Xiang and B. 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. |
[25] |
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49, American Mathematical Society, Providence, RI, 1997.
doi: 10.1090/surv/049. |
[26] |
H. Song and D. Xue,
Blow up in a nonlinear viscoelastic wave equation with strong damping, Nonlinear Anal., 109 (2014), 245-251.
doi: 10.1016/j.na.2014.06.012. |
[27] |
J. L. Vázquez, Nonlinear diffusion with fractional Laplacian operators, in Nonlinear Partial Differential Equations, Abel Symp., 7, Springer, Heidelberg, 2012,271–298.
doi: 10.1007/978-3-642-25361-4_15. |
[28] |
F. Wang, D. Hu and M. Xiang,
Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems, Adv. Nonlinear Anal., 10 (2021), 636-658.
doi: 10.1515/anona-2020-0150. |
[29] |
X. Wang and R. Xu,
Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation, Adv. Nonlinear Anal., 10 (2021), 261-288.
doi: 10.1515/anona-2020-0141. |
[30] |
M. Xiang, G. M. Bisci and B. Zhang,
Variational analysis for nonlocal Yamabe-type systems, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2069-2094.
doi: 10.3934/dcdss.2020159. |
[31] |
M. Xiang, D. Hu, B. Zhang and Y. Wang, Multiplicity of solutions for variable-order fractional Kirchhoff equations with nonstandard growth, J. Math. Anal. Appl., 501 (2021), 19pp.
doi: 10.1016/j.jmaa.2020.124269. |
[32] |
M. Xiang, V. D. Rǎdulescu and B. Zhang, A critical fractional Choquard-Kirchhoff problem with magnetic field, Commun. Contemp. Math., 21 (2019), 36pp.
doi: 10.1142/s0219199718500049. |
[33] |
M. Xiang, V. D. Rǎdulescu and B. Zhang, Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity, Cal. Var. Partial Differential Equations, 58 (2019), 27pp.
doi: 10.1007/s00526-019-1499-y. |
[34] |
M. Xiang, V. D. Rǎdulescu and B. Zhang,
Nonlocal Kirchhoff diffusion problems: Local existence and blow-up of solutions, Nonlinearity, 31 (2018), 3228-3250.
doi: 10.1088/1361-6544/aaba35. |
[35] |
M. Xiang, B. Zhang and M. Ferrara,
Existence of solutions for Kirchhoff type problem involving the non-local fractional $p$-Laplacian, J. Math. Anal. Appl., 424 (2015), 1021-1041.
doi: 10.1016/j.jmaa.2014.11.055. |
[36] |
M. Xiang, B. Zhang and D. Hu,
Kirchhoff-type differential inclusion problems involving the fractional Laplacian and strong damping, Electron. Res. Arch., 28 (2020), 651-669.
doi: 10.3934/era.2020034. |
[37] |
R. Xu, Y. Yang and Y. Liu,
Global well-posedness for strongly damped viscoelastic wave equation, Appl. Anal., 92 (2013), 138-157.
doi: 10.1080/00036811.2011.601456. |
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