November  2021, 20(11): 3703-3721. doi: 10.3934/cpaa.2021127

A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime

1. 

Basic Sciences Department, Deanship of Preparatory Year and Supporting Studies, P. O. Box 1982, Imam Abdulrahman Bin Faisal University, Dammam, KSA

2. 

Department of Mathematics, University of Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy

* Corresponding author

Received  February 2021 Revised  June 2021 Published  November 2021 Early access  July 2021

Fund Project: The third author is supported by the GNAMPA project 'Problemi stazionari e di evoluzione nelle equazioni di campo nonlineari dispersive'

In this paper, we establish blow-up results for the semilinear wave equation in generalized Einstein-de Sitter spacetime with nonlinearity of derivative type. Our approach is based on the integral representation formula for the solution to the corresponding linear problem in the one-dimensional case, that we will determine through Yagdjian's Integral Transform approach. As upper bound for the exponent of the nonlinear term, we discover a Glassey-type exponent which depends both on the space dimension and on the Lorentzian metric in the generalized Einstein-de Sitter spacetime.

Citation: Makram Hamouda, Mohamed Ali Hamza, Alessandro Palmieri. A note on the nonexistence of global solutions to the semilinear wave equation with nonlinearity of derivative-type in the generalized Einstein-de Sitter spacetime. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3703-3721. doi: 10.3934/cpaa.2021127
References:
[1]

M. D'Abbicco, Small data solutions for the Euler-Poisson-Darboux equation with a power nonlinearity, J. Differ. Equ., 286 (2021), 531-556.  doi: 10.1016/j.jde.2021.03.033.  Google Scholar

[2]

A. Galstian, T. Kinoshita and K. Yagdjian, A note on wave equation in Einstein and de Sitter space-time, J. Math. Phys., 51 (2010), 052501. doi: 10.1063/1.3387249.  Google Scholar

[3]

A. Galstian and K. Yagdjian, Finite lifespan of solutions of the semilinear wave equation in the Einstein-de Sitter spacetime, Rev. Math. Phys., 32 (2020), 2050018. doi: 10.1142/S0129055X2050018X.  Google Scholar

[4]

M. Hamouda and M. A. Hamza, Blow-up for wave equation with the scale-invariant damping and combined nonlinearities, Math. Methods Appl. Sci., 44 (2021) 1127-1136. doi: 10.1002/mma. 6817.  Google Scholar

[5]

M. Hamouda and M. A. Hamza, Improvement on the blow-up of the wave equation with the scale-invariant damping and combined nonlinearities, Nonlinear Anal. Real World Appl., 59 (2021), 103275. doi: 10.1016/j. nonrwa. 2020.103275.  Google Scholar

[6]

M. Hamouda and M. A. Hamza, Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities, preprint, arXiv: 2011.04895. Google Scholar

[7]

M. Hamouda, M. A. Hamza and A. Palmieri, Blow-up and lifespan estimates for a damped wave equation in the Einstein-de Sitter spacetime with nonlinearity of derivative type, arXiv: 2102.01137. Google Scholar

[8]

K. HidanoC. Wang and K. Yokoyama, The Glassey conjecture with radially symmetric data, J. Math. Pures Appl., 98 (2012), 518-541.  doi: 10.1016/j.matpur.2012.01.007.  Google Scholar

[9]

N. A. Lai and N. M. Schiavone, Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture, preprint, arXiv: 2007.16003v2. Google Scholar

[10]

S. Lucente and A. Palmieri, A blow-up result for a generalized Tricomi equation with nonlinearity of derivative type, Milan J. Math., 89 (2021), 45-57.  doi: 10.1007/s00032-021-00326-x.  Google Scholar

[11]

W. Nunes do NascimentoA. Palmieri and M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, Math. Nachr., 290 (2017), 1779-1805.  doi: 10.1002/mana.201600069.  Google Scholar

[12] F. W. J. OlverD. W. LozierR. F. Boisvert and C. W. Clark, NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010.   Google Scholar
[13]

A. Palmieri, Integral representation formulae for the solution of a wave equation with time-dependent damping and mass in the scale-invariant case, Math. Methods Appl. Sci., (2021), 1-32. doi: 10.1002/mma. 7603.  Google Scholar

[14]

A. Palmieri, Lifespan estimates for local solutions to the semilinear wave equation in Einstein-de Sitter spacetime, preprint, arXiv: 2009.04388. Google Scholar

[15]

A. Palmieri, Blow-up results for semilinear damped wave equations in Einstein-de Sitter spacetime, Z. Angew. Math. Phys., 72 (2021), 64. doi: 10.1007/s00033-021-01494-x.  Google Scholar

[16]

A. Palmieri and M. Reissig, A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass, J. Differ. Equ., 266 (2019), 1176-1220.  doi: 10.1016/j.jde.2018.07.061.  Google Scholar

[17]

A. Palmieri and Z. Tu, Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity, J. Math. Anal. Appl., 470 (2019), 447-469.  doi: 10.1016/j.jmaa.2018.10.015.  Google Scholar

[18]

A. Palmieri and Z. Tu, A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type, Calc. Var. Partial Differ. Equ. 60 (2021), 72. doi: 10.1007/s00526-021-01948-0.  Google Scholar

[19]

K. Tsutaya and Y. Wakasugi, Blow up of solutions of semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime, J. Math. Phys., 61 (2020), 091503. doi: 10.1063/1.5139301.  Google Scholar

[20]

K. Tsutaya and Y. Wakasugi, On heatlike lifespan of solutions of semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime, J. Math. Anal. Appl., 500 (2021), 125133. doi: 10.1016/j. jmaa. 2021.125133.  Google Scholar

[21]

N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac system and wave equation with small data, Tsukuba J. Math., 22 (1998), 193-211.  doi: 10.21099/tkbjm/1496163480.  Google Scholar

[22]

K. Yagdjian, A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain, J. Differ. Equ., 206 (2004), 227-252.  doi: 10.1016/j.jde.2004.07.028.  Google Scholar

[23]

K. Yagdjian, The self-similar solutions of the Tricomi-type equations, Z. Angew. Math. Phys., 58 (2007), 612-645.  doi: 10.1007/s00033-006-5099-2.  Google Scholar

[24]

K. Yagdjian, Fundamental solutions for hyperbolic operators with variable coefficients, Rend. Istit. Mat. Univ. Trieste, 42 (2010), 221-243.   Google Scholar

[25]

K. Yagdjian, Integral transform approach to generalized Tricomi equations, J. Differ. Equ., 259 (2015), 5927-5981.  doi: 10.1016/j.jde.2015.07.014.  Google Scholar

[26]

K. Yagdjian, Fundamental solutions of the Dirac operator in the Friedmann-Lemaître-Robertson-Walker spacetime, Ann. Phys., 421 (2020), 168266. doi: 10.1016/j. aop. 2020.168266.  Google Scholar

[27]

K. Yagdjian and A. Galstian, Fundamental solutions of the wave equation in Robertson-Walker spaces, J. Math. Anal. Appl., 346 (2008), 501-520.  doi: 10.1016/j.jmaa.2008.05.075.  Google Scholar

[28]

K. Yagdjian and A. Galstian, Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime, Commun. Math. Phys., 285 (2009), 293-344.  doi: 10.1007/s00220-008-0649-4.  Google Scholar

[29]

Y. Zhou, Blow up of solutions to the Cauchy problem for nonlinear wave equations, Chin. Ann. Math. Ser. B, 22 (2001), 275-280.  doi: 10.1142/S0252959901000280.  Google Scholar

show all references

References:
[1]

M. D'Abbicco, Small data solutions for the Euler-Poisson-Darboux equation with a power nonlinearity, J. Differ. Equ., 286 (2021), 531-556.  doi: 10.1016/j.jde.2021.03.033.  Google Scholar

[2]

A. Galstian, T. Kinoshita and K. Yagdjian, A note on wave equation in Einstein and de Sitter space-time, J. Math. Phys., 51 (2010), 052501. doi: 10.1063/1.3387249.  Google Scholar

[3]

A. Galstian and K. Yagdjian, Finite lifespan of solutions of the semilinear wave equation in the Einstein-de Sitter spacetime, Rev. Math. Phys., 32 (2020), 2050018. doi: 10.1142/S0129055X2050018X.  Google Scholar

[4]

M. Hamouda and M. A. Hamza, Blow-up for wave equation with the scale-invariant damping and combined nonlinearities, Math. Methods Appl. Sci., 44 (2021) 1127-1136. doi: 10.1002/mma. 6817.  Google Scholar

[5]

M. Hamouda and M. A. Hamza, Improvement on the blow-up of the wave equation with the scale-invariant damping and combined nonlinearities, Nonlinear Anal. Real World Appl., 59 (2021), 103275. doi: 10.1016/j. nonrwa. 2020.103275.  Google Scholar

[6]

M. Hamouda and M. A. Hamza, Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities, preprint, arXiv: 2011.04895. Google Scholar

[7]

M. Hamouda, M. A. Hamza and A. Palmieri, Blow-up and lifespan estimates for a damped wave equation in the Einstein-de Sitter spacetime with nonlinearity of derivative type, arXiv: 2102.01137. Google Scholar

[8]

K. HidanoC. Wang and K. Yokoyama, The Glassey conjecture with radially symmetric data, J. Math. Pures Appl., 98 (2012), 518-541.  doi: 10.1016/j.matpur.2012.01.007.  Google Scholar

[9]

N. A. Lai and N. M. Schiavone, Blow-up and lifespan estimate for generalized Tricomi equations related to Glassey conjecture, preprint, arXiv: 2007.16003v2. Google Scholar

[10]

S. Lucente and A. Palmieri, A blow-up result for a generalized Tricomi equation with nonlinearity of derivative type, Milan J. Math., 89 (2021), 45-57.  doi: 10.1007/s00032-021-00326-x.  Google Scholar

[11]

W. Nunes do NascimentoA. Palmieri and M. Reissig, Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, Math. Nachr., 290 (2017), 1779-1805.  doi: 10.1002/mana.201600069.  Google Scholar

[12] F. W. J. OlverD. W. LozierR. F. Boisvert and C. W. Clark, NIST handbook of mathematical functions, U.S. Department of Commerce, National Institute of Standards and Technology, Washington, DC; Cambridge University Press, Cambridge, 2010.   Google Scholar
[13]

A. Palmieri, Integral representation formulae for the solution of a wave equation with time-dependent damping and mass in the scale-invariant case, Math. Methods Appl. Sci., (2021), 1-32. doi: 10.1002/mma. 7603.  Google Scholar

[14]

A. Palmieri, Lifespan estimates for local solutions to the semilinear wave equation in Einstein-de Sitter spacetime, preprint, arXiv: 2009.04388. Google Scholar

[15]

A. Palmieri, Blow-up results for semilinear damped wave equations in Einstein-de Sitter spacetime, Z. Angew. Math. Phys., 72 (2021), 64. doi: 10.1007/s00033-021-01494-x.  Google Scholar

[16]

A. Palmieri and M. Reissig, A competition between Fujita and Strauss type exponents for blow-up of semi-linear wave equations with scale-invariant damping and mass, J. Differ. Equ., 266 (2019), 1176-1220.  doi: 10.1016/j.jde.2018.07.061.  Google Scholar

[17]

A. Palmieri and Z. Tu, Lifespan of semilinear wave equation with scale invariant dissipation and mass and sub-Strauss power nonlinearity, J. Math. Anal. Appl., 470 (2019), 447-469.  doi: 10.1016/j.jmaa.2018.10.015.  Google Scholar

[18]

A. Palmieri and Z. Tu, A blow-up result for a semilinear wave equation with scale-invariant damping and mass and nonlinearity of derivative type, Calc. Var. Partial Differ. Equ. 60 (2021), 72. doi: 10.1007/s00526-021-01948-0.  Google Scholar

[19]

K. Tsutaya and Y. Wakasugi, Blow up of solutions of semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime, J. Math. Phys., 61 (2020), 091503. doi: 10.1063/1.5139301.  Google Scholar

[20]

K. Tsutaya and Y. Wakasugi, On heatlike lifespan of solutions of semilinear wave equations in Friedmann-Lemaître-Robertson-Walker spacetime, J. Math. Anal. Appl., 500 (2021), 125133. doi: 10.1016/j. jmaa. 2021.125133.  Google Scholar

[21]

N. Tzvetkov, Existence of global solutions to nonlinear massless Dirac system and wave equation with small data, Tsukuba J. Math., 22 (1998), 193-211.  doi: 10.21099/tkbjm/1496163480.  Google Scholar

[22]

K. Yagdjian, A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain, J. Differ. Equ., 206 (2004), 227-252.  doi: 10.1016/j.jde.2004.07.028.  Google Scholar

[23]

K. Yagdjian, The self-similar solutions of the Tricomi-type equations, Z. Angew. Math. Phys., 58 (2007), 612-645.  doi: 10.1007/s00033-006-5099-2.  Google Scholar

[24]

K. Yagdjian, Fundamental solutions for hyperbolic operators with variable coefficients, Rend. Istit. Mat. Univ. Trieste, 42 (2010), 221-243.   Google Scholar

[25]

K. Yagdjian, Integral transform approach to generalized Tricomi equations, J. Differ. Equ., 259 (2015), 5927-5981.  doi: 10.1016/j.jde.2015.07.014.  Google Scholar

[26]

K. Yagdjian, Fundamental solutions of the Dirac operator in the Friedmann-Lemaître-Robertson-Walker spacetime, Ann. Phys., 421 (2020), 168266. doi: 10.1016/j. aop. 2020.168266.  Google Scholar

[27]

K. Yagdjian and A. Galstian, Fundamental solutions of the wave equation in Robertson-Walker spaces, J. Math. Anal. Appl., 346 (2008), 501-520.  doi: 10.1016/j.jmaa.2008.05.075.  Google Scholar

[28]

K. Yagdjian and A. Galstian, Fundamental Solutions for the Klein-Gordon Equation in de Sitter Spacetime, Commun. Math. Phys., 285 (2009), 293-344.  doi: 10.1007/s00220-008-0649-4.  Google Scholar

[29]

Y. Zhou, Blow up of solutions to the Cauchy problem for nonlinear wave equations, Chin. Ann. Math. Ser. B, 22 (2001), 275-280.  doi: 10.1142/S0252959901000280.  Google Scholar

[1]

Makoto Nakamura. Remarks on a dispersive equation in de Sitter spacetime. Conference Publications, 2015, 2015 (special) : 901-905. doi: 10.3934/proc.2015.0901

[2]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679

[3]

Wenhui Chen, Alessandro Palmieri. A blow – up result for the semilinear Moore – Gibson – Thompson equation with nonlinearity of derivative type in the conservative case. Evolution Equations & Control Theory, 2021, 10 (4) : 673-687. doi: 10.3934/eect.2020085

[4]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069

[5]

Xiaoliang Li, Baiyu Liu. Finite time blow-up and global solutions for a nonlocal parabolic equation with Hartree type nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 3093-3112. doi: 10.3934/cpaa.2020134

[6]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388

[7]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[8]

Helin Guo, Yimin Zhang, Huansong Zhou. Blow-up solutions for a Kirchhoff type elliptic equation with trapping potential. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1875-1897. doi: 10.3934/cpaa.2018089

[9]

Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4847-4885. doi: 10.3934/dcds.2021060

[10]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042

[11]

Li Ma. Blow-up for semilinear parabolic equations with critical Sobolev exponent. Communications on Pure & Applied Analysis, 2013, 12 (2) : 1103-1110. doi: 10.3934/cpaa.2013.12.1103

[12]

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

[13]

Yoshikazu Giga. Interior derivative blow-up for quasilinear parabolic equations. Discrete & Continuous Dynamical Systems, 1995, 1 (3) : 449-461. doi: 10.3934/dcds.1995.1.449

[14]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831

[15]

Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280

[16]

Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108

[17]

Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021011

[18]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006

[19]

Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021093

[20]

Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete & Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (112)
  • HTML views (196)
  • Cited by (0)

[Back to Top]