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

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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 and Continuous Dynamical Systems - S, 2021, 14 (12) : 4609-4629. doi: 10.3934/dcdss.2021125

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.
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[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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