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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

Makram Hamouda 1, , Mohamed Ali Hamza 1, and  Alessandro Palmieri 2,,

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.

Keywords: Wave equation, Einstein-de Sitter spacetime, Integral transform, Blow-up, Glassey exponent, Nonlinearity of derivative type.
Mathematics Subject Classification: Primary: 35B44, 35C15, 35L71; Secondary: 35A08, 35B33, 35L15.
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. Hidano, C. 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 Nascimento, A. 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. Olver, D. W. Lozier, R. 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. Hidano, C. 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 Nascimento, A. 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. Olver, D. W. Lozier, R. 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

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