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December  2021, 14(12): 4609-4629. doi: 10.3934/dcdss.2021125

Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity

Mingqi Xiang , and  Die Hu

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

* Corresponding author: Mingqi Xiang

Received  August 2021 Revised  September 2021 Published  December 2021 Early access  October 2021

In this paper, we deal with the initial boundary value problem of the following fractional wave equation of Kirchhoff type
$ \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*} $
where
$ M:[0, \infty)\rightarrow (0, \infty) $
 is a nondecreasing and continuous function,
$ [u]_{\alpha, 2} $
 is the Gagliardo-seminorm of
$ u $
,
$ (-\Delta)^\alpha $
 and
$ (-\Delta)^s $
 are the fractional Laplace operators,
$ g:\mathbb{R}^+\rightarrow \mathbb{R}^+ $
 is a positive nonincreasing function and
$ \lambda $
 is a parameter. First, the local and global existence of solutions are obtained by using the Galerkin method. Then the global nonexistence of solutions is discussed via blow-up analysis. Our results generalize and improve the existing results in the literature.
Keywords: Fractional Laplacian, Galerkin approximation, blow-up, viscoelastic, Kirchhoff problems.
Mathematics Subject Classification: 35L20, 35B44, 35D30, 47G20.
Citation: Mingqi Xiang, Die Hu. Existence and blow-up of solutions for fractional wave equations of Kirchhoff type with viscoelasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (12) : 4609-4629. doi: 10.3934/dcdss.2021125
References:
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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.  Google Scholar

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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.  Google Scholar

[9]

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[10]

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[11]

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[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.  Google Scholar

[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.  Google Scholar

[17]

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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.  Google Scholar

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Y. Liu, Long-time behavior of a class of viscoelastic plate equations, Electron. Res. Arch., 28 (2020), 311-326.  doi: 10.3934/era.2020018.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[25]

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[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

P. Hartman, Ordinary Differential Equations, Birkhäuser, Boston, Mass., 1982.  Google Scholar

[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.  Google Scholar

[12]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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