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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 |
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.
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
M. Hamouda and M. A. Hamza, Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities, preprint, arXiv: 2011.04895. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
![]() ![]() |
[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. |
[14] |
A. Palmieri, Lifespan estimates for local solutions to the semilinear wave equation in Einstein-de Sitter spacetime, preprint, arXiv: 2009.04388. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
K. Yagdjian,
Fundamental solutions for hyperbolic operators with variable coefficients, Rend. Istit. Mat. Univ. Trieste, 42 (2010), 221-243.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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. |
[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. |
[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. |
[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. |
[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. |
[6] |
M. Hamouda and M. A. Hamza, Blow-up and lifespan estimate for the generalized Tricomi equation with mixed nonlinearities, preprint, arXiv: 2011.04895. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
![]() ![]() |
[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. |
[14] |
A. Palmieri, Lifespan estimates for local solutions to the semilinear wave equation in Einstein-de Sitter spacetime, preprint, arXiv: 2009.04388. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[24] |
K. Yagdjian,
Fundamental solutions for hyperbolic operators with variable coefficients, Rend. Istit. Mat. Univ. Trieste, 42 (2010), 221-243.
|
[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. |
[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. |
[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. |
[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. |
[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. |
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