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doi: 10.3934/mcrf.2022022
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Lifespan estimates of solutions to quasilinear wave equations with damping and negative mass term

1. 

Data Science and Technology, North University of China, Taiyuan 030051, China

2. 

Department of Mathematics, North University of China, Taiyuan 030051, China

*Corresponding author

Received  November 2021 Early access May 2022

Fund Project: The project is supported by Fundamental Research Program of Shanxi Province(No. 20210302123021, No. 20210302123045), Innovative Research Team of North University of China (No. TD201901), Program for the Innovative Talents of Higher Education Institutions of Shanxi Province

The main goal of this work is to investigate formation of singularities for solutions to the quasilinear wave equations with damping terms, negative mass terms and divergence form nonlinearities in the critical and sub-critical cases. Upper bound lifespan estimates of solutions are derived by applying the rescaled test function method and iteration technique. The results are the same as corresponding wave equation without damping term and mass term. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent. To the best of our knowledge, the results in Theorems $ 1.1-1.4 $ are new. Moreover, the changing trends of semilinear wave equations are illustrated through numerical simulation.

Citation: Jie Yang, Sen Ming, Wei Han, Xiongmei Fan. Lifespan estimates of solutions to quasilinear wave equations with damping and negative mass term. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022022
References:
[1]

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

[2]

W. DaiD. Y. Fang and C. B. Wang, Global existence and lifespan for semilinear wave equations with mixed nonlinear terms, J. Diff. Equa., 267 (2019), 3328-3354.  doi: 10.1016/j.jde.2019.04.007.

[3]

T. A. Dao and A. Z. Fino, Critical exponent for semilinear structurally damped wave equation of derivative type, Math. Meth. Appl. Scie., 43 (2020), 9766-9775.  doi: 10.1002/mma.6649.

[4]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t} = \Delta t+u^{1+\alpha}$, J. Facu. Scie. Univ. Toky. Sect., 13 (1966), 109-124. 

[5]

R. T. Glassey, Finite time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066.

[6]

R. T. Glassey, Existence in the large for $u_tt-\Delta u = F(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042.

[7]

R. T. Glassey, Math Review to Global behavior of solutions to nonlinear wave equations in three space dimensions, Comm. Part. Diff. Equa., 8 (1983), 1291-1323. 

[8]

M. Hamouda and M. A. Hamza, Blow-up for wave equation with the scale invariant damping and combined nonlinearities, Math. Meth. Appl. Scie., 44 (2020), 1127-1136.  doi: 10.1002/mma.6817.

[9]

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

[10]

W. Han, The rescaling method for some critical quasilinear wave equations with the divergence form of the nonlinearity, Appl. Anal., 98 (2019), 2525-2544.  doi: 10.1080/00036811.2018.1466280.

[11]

M. IkedaT. Tanaka and K. Wakasa, Critical exponent for the wave equation with a time dependent scale invariant damping and a cubic convolution, J. Diff. Equa., 270 (2021), 916-946.  doi: 10.1016/j.jde.2020.08.047.

[12]

M. IkedaZ. H. Tu and K. Wakasa, Small data blow-up of semilinear wave equation with scattering dissipation and time dependent mass, Evol. Equ. Control Theory, 11 (2022), 515-536.  doi: 10.3934/eect.2021011.

[13]

T. ImaiM. KatoH. Takamura and K. Wakasa, The lifespan of solutions of semilinear wave equations with the scale invariant damping in two space dimensions, J. Diff. Equa., 269 (2020), 8387-8424.  doi: 10.1016/j.jde.2020.06.019.

[14]

F. John, Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.  doi: 10.1002/cpa.3160340103.

[15]

N. A. LaiM. Y. LiuK. Wakasa and C. B. Wang, Lifespan estimates for $2$ dimensional semilinear wave equations in asymptotically Euclidean exterior domains, J. Func. Anal., 281 (2021), 109253.  doi: 10.1016/j.jfa.2021.109253.

[16]

N. A. Lai, M. Y. Liu, Z. H. Tu and C. B. Wang, Lifespan estimates for semilinear wave equations with sapce dependent damping and potential, arXiv: 2102.10257v1, 2021.

[17]

N. A. Lai, N. M. Schiavone and H. Takamura, Wave like blow-up for semilinear wave equations with scattering damping and negative mass term, New Tool. Nonl. PDEs Appl., (2019), 217–240. doi: 10.1007/978-3-030-10937-0_8.

[18]

N. A. LaiN. M. Schiavone and H. Takamura, Heat like and wave like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma, J. Diff. Equ., 269 (2020), 11575-11620.  doi: 10.1016/j.jde.2020.08.020.

[19]

N. A. Lai and H. Takamura, Blow-up for semilinear damped wave equations with sub-critical exponent in the scattering case, Nonl. Anal., 168 (2018), 222-237.  doi: 10.1016/j.na.2017.12.008.

[20]

N. A. Lai and H. Takamura., Non-existence of global solutions of nonlinear wave equations with weak time dependent damping related to Glassey's conjecture, Diff. Inte. Equa., 32 (2019), 37-48. 

[21]

N. A. Lai and H. Takamura, Non-existence of global solutions of wave equations with weak time dependent damping and combined nonlinearity, Nonl. Anal. Real Worl. Appl., 45 (2019), 83-96.  doi: 10.1016/j.nonrwa.2018.06.008.

[22]

N. A. Lai and Z. H. Tu, Strauss exponent for semilinear wave equations with scsttering space dependent damping, J. Math. Anal. Appl., 489 (2020), 124189.  doi: 10.1016/j.jmaa.2020.124189.

[23]

Q. Lei and H. Yang, Global existence and blow-up for semilinear wave equations with variable coefficients, Chin. Anna. Math. Seri. B., 39 (2018), 643-664.  doi: 10.1007/s11401-018-0087-3.

[24]

T. T. Li and Y. Zhou, Breakdown of solutions to $u_tt+u_{t} = u^{1+\alpha}$, Disc. Cont. dyna. Syst., 1 (1995), 503-520.  doi: 10.3934/dcds.1995.1.503.

[25]

Y. H. LinN. A. Lai and S. Ming, Lifespan estimate for semilinear wave equation in Schwarzschild spacetime, Appl. Math. Lett., 99 (2020), 105997.  doi: 10.1016/j.aml.2019.105997.

[26]

H. Lindblad, Blow up for solutions of $u_tt-\Delta u = |u|^{p}$ with small initial data, Comm. Part. Diff. Equa., 15 (1990), 757-821.  doi: 10.1080/03605309908820708.

[27]

M. Y. Liu and C. B. Wang, Blow up for small amplitude semilinear wave equations with mixed nonlinearities on asymptotically Euclidean manifolds, J. Diff. Equa., 269 (2020), 8573-8596.  doi: 10.1016/j.jde.2020.06.032.

[28]

S. MingS. Y. Lai and X. M. Fan, Lifespan estimates of solutions to quasilinear wave equations with scattering damping, J. Math. Anal. Appl., 492 (2020), 124441.  doi: 10.1016/j.jmaa.2020.124441.

[29]

S. Ming, S. Y. Lai and X. M. Fan, Blow-up for a coupled system of semilinear wave equations with scattering dampings and combined nonlinearities, Appl. Anal., 2020. doi: 10.1016/j.jmaa.2020.124189.

[30]

A. Palmieri and Z. H. Tu, A blow-up result for a semilinear wave equation with scale invariant damping and mass and nonlinearity of derivative type, Calc. Vari. Part. Diff. Equa., 6 (2021), Paper No. 72, 23 pp. doi: 10.1007/s00526-021-01948-0.

[31]

T. C. Sideris, Non-existence of global solutions to semilinear wave equations in high dimensions, J. Diff. Equa., 52 (1984), 378-406.  doi: 10.1016/0022-0396(84)90169-4.

[32]

K. Wakasa and B. Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients, J. Diff. Equa., 266 (2019), 5360-5376.  doi: 10.1016/j.jde.2018.10.028.

[33]

K. Wakasa and B. Yordanov, On the blow-up for critical semilinear wave equations with damping in the scattering case, Nonl. Anal., 180 (2019), 67-74.  doi: 10.1016/j.na.2018.09.012.

[34]

C. B. Wang and X. Yu, Recent works on the Strauss conjecture, Rece. Adva. Harm. Anal. Part. Diff. Equa., 581 (2012), 235-256.  doi: 10.1090/conm/581/11497.

[35]

J. Wirth, Solution representations for a wave equation with weak dissipation, Math. Meth. Appl. Scie., 27 (2004), 101-124.  doi: 10.1002/mma.446.

[36]

J. Wirth, Wave equations with time dependent dissipation. Ⅰ. noneffective dissipation, J. Diff. Equa., 222 (2006), 487-514.  doi: 10.1016/j.jde.2005.07.019.

[37]

J. Wirth, Wave equations with time dependent dissipation. Ⅱ. Effective dissipation, J. Diff. Equa., 232 (2007), 74-103.  doi: 10.1016/j.jde.2006.06.004.

[38]

B. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Func. Anal., 231 (2006), 361-374.  doi: 10.1016/j.jfa.2005.03.012.

[39]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, Comp. Rend. Acad. Scie. Pari. Séri. I Math., 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.

[40]

Y. Zhou, Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Part. Diff. Equa., 8 (1995), 135-144. 

[41]

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

[42]

Y. Zhou, Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chin. Annu. Math. Seri. B, 28 (2007), 205-212.  doi: 10.1007/s11401-005-0205-x.

[43]

Y. Zhou and W. Han, Lifespan of solutions to critical semilinear wave equations, Comm. Part. Diff. Equa., 39 (2014), 439-451.  doi: 10.1080/03605302.2013.863914.

show all references

References:
[1]

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

[2]

W. DaiD. Y. Fang and C. B. Wang, Global existence and lifespan for semilinear wave equations with mixed nonlinear terms, J. Diff. Equa., 267 (2019), 3328-3354.  doi: 10.1016/j.jde.2019.04.007.

[3]

T. A. Dao and A. Z. Fino, Critical exponent for semilinear structurally damped wave equation of derivative type, Math. Meth. Appl. Scie., 43 (2020), 9766-9775.  doi: 10.1002/mma.6649.

[4]

H. Fujita, On the blowing up of solutions of the Cauchy problem for $u_{t} = \Delta t+u^{1+\alpha}$, J. Facu. Scie. Univ. Toky. Sect., 13 (1966), 109-124. 

[5]

R. T. Glassey, Finite time blow-up for solutions of nonlinear wave equations, Math. Z., 177 (1981), 323-340.  doi: 10.1007/BF01162066.

[6]

R. T. Glassey, Existence in the large for $u_tt-\Delta u = F(u)$ in two space dimensions, Math. Z., 178 (1981), 233-261.  doi: 10.1007/BF01262042.

[7]

R. T. Glassey, Math Review to Global behavior of solutions to nonlinear wave equations in three space dimensions, Comm. Part. Diff. Equa., 8 (1983), 1291-1323. 

[8]

M. Hamouda and M. A. Hamza, Blow-up for wave equation with the scale invariant damping and combined nonlinearities, Math. Meth. Appl. Scie., 44 (2020), 1127-1136.  doi: 10.1002/mma.6817.

[9]

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

[10]

W. Han, The rescaling method for some critical quasilinear wave equations with the divergence form of the nonlinearity, Appl. Anal., 98 (2019), 2525-2544.  doi: 10.1080/00036811.2018.1466280.

[11]

M. IkedaT. Tanaka and K. Wakasa, Critical exponent for the wave equation with a time dependent scale invariant damping and a cubic convolution, J. Diff. Equa., 270 (2021), 916-946.  doi: 10.1016/j.jde.2020.08.047.

[12]

M. IkedaZ. H. Tu and K. Wakasa, Small data blow-up of semilinear wave equation with scattering dissipation and time dependent mass, Evol. Equ. Control Theory, 11 (2022), 515-536.  doi: 10.3934/eect.2021011.

[13]

T. ImaiM. KatoH. Takamura and K. Wakasa, The lifespan of solutions of semilinear wave equations with the scale invariant damping in two space dimensions, J. Diff. Equa., 269 (2020), 8387-8424.  doi: 10.1016/j.jde.2020.06.019.

[14]

F. John, Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math., 34 (1981), 29-51.  doi: 10.1002/cpa.3160340103.

[15]

N. A. LaiM. Y. LiuK. Wakasa and C. B. Wang, Lifespan estimates for $2$ dimensional semilinear wave equations in asymptotically Euclidean exterior domains, J. Func. Anal., 281 (2021), 109253.  doi: 10.1016/j.jfa.2021.109253.

[16]

N. A. Lai, M. Y. Liu, Z. H. Tu and C. B. Wang, Lifespan estimates for semilinear wave equations with sapce dependent damping and potential, arXiv: 2102.10257v1, 2021.

[17]

N. A. Lai, N. M. Schiavone and H. Takamura, Wave like blow-up for semilinear wave equations with scattering damping and negative mass term, New Tool. Nonl. PDEs Appl., (2019), 217–240. doi: 10.1007/978-3-030-10937-0_8.

[18]

N. A. LaiN. M. Schiavone and H. Takamura, Heat like and wave like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma, J. Diff. Equ., 269 (2020), 11575-11620.  doi: 10.1016/j.jde.2020.08.020.

[19]

N. A. Lai and H. Takamura, Blow-up for semilinear damped wave equations with sub-critical exponent in the scattering case, Nonl. Anal., 168 (2018), 222-237.  doi: 10.1016/j.na.2017.12.008.

[20]

N. A. Lai and H. Takamura., Non-existence of global solutions of nonlinear wave equations with weak time dependent damping related to Glassey's conjecture, Diff. Inte. Equa., 32 (2019), 37-48. 

[21]

N. A. Lai and H. Takamura, Non-existence of global solutions of wave equations with weak time dependent damping and combined nonlinearity, Nonl. Anal. Real Worl. Appl., 45 (2019), 83-96.  doi: 10.1016/j.nonrwa.2018.06.008.

[22]

N. A. Lai and Z. H. Tu, Strauss exponent for semilinear wave equations with scsttering space dependent damping, J. Math. Anal. Appl., 489 (2020), 124189.  doi: 10.1016/j.jmaa.2020.124189.

[23]

Q. Lei and H. Yang, Global existence and blow-up for semilinear wave equations with variable coefficients, Chin. Anna. Math. Seri. B., 39 (2018), 643-664.  doi: 10.1007/s11401-018-0087-3.

[24]

T. T. Li and Y. Zhou, Breakdown of solutions to $u_tt+u_{t} = u^{1+\alpha}$, Disc. Cont. dyna. Syst., 1 (1995), 503-520.  doi: 10.3934/dcds.1995.1.503.

[25]

Y. H. LinN. A. Lai and S. Ming, Lifespan estimate for semilinear wave equation in Schwarzschild spacetime, Appl. Math. Lett., 99 (2020), 105997.  doi: 10.1016/j.aml.2019.105997.

[26]

H. Lindblad, Blow up for solutions of $u_tt-\Delta u = |u|^{p}$ with small initial data, Comm. Part. Diff. Equa., 15 (1990), 757-821.  doi: 10.1080/03605309908820708.

[27]

M. Y. Liu and C. B. Wang, Blow up for small amplitude semilinear wave equations with mixed nonlinearities on asymptotically Euclidean manifolds, J. Diff. Equa., 269 (2020), 8573-8596.  doi: 10.1016/j.jde.2020.06.032.

[28]

S. MingS. Y. Lai and X. M. Fan, Lifespan estimates of solutions to quasilinear wave equations with scattering damping, J. Math. Anal. Appl., 492 (2020), 124441.  doi: 10.1016/j.jmaa.2020.124441.

[29]

S. Ming, S. Y. Lai and X. M. Fan, Blow-up for a coupled system of semilinear wave equations with scattering dampings and combined nonlinearities, Appl. Anal., 2020. doi: 10.1016/j.jmaa.2020.124189.

[30]

A. Palmieri and Z. H. Tu, A blow-up result for a semilinear wave equation with scale invariant damping and mass and nonlinearity of derivative type, Calc. Vari. Part. Diff. Equa., 6 (2021), Paper No. 72, 23 pp. doi: 10.1007/s00526-021-01948-0.

[31]

T. C. Sideris, Non-existence of global solutions to semilinear wave equations in high dimensions, J. Diff. Equa., 52 (1984), 378-406.  doi: 10.1016/0022-0396(84)90169-4.

[32]

K. Wakasa and B. Yordanov, Blow-up of solutions to critical semilinear wave equations with variable coefficients, J. Diff. Equa., 266 (2019), 5360-5376.  doi: 10.1016/j.jde.2018.10.028.

[33]

K. Wakasa and B. Yordanov, On the blow-up for critical semilinear wave equations with damping in the scattering case, Nonl. Anal., 180 (2019), 67-74.  doi: 10.1016/j.na.2018.09.012.

[34]

C. B. Wang and X. Yu, Recent works on the Strauss conjecture, Rece. Adva. Harm. Anal. Part. Diff. Equa., 581 (2012), 235-256.  doi: 10.1090/conm/581/11497.

[35]

J. Wirth, Solution representations for a wave equation with weak dissipation, Math. Meth. Appl. Scie., 27 (2004), 101-124.  doi: 10.1002/mma.446.

[36]

J. Wirth, Wave equations with time dependent dissipation. Ⅰ. noneffective dissipation, J. Diff. Equa., 222 (2006), 487-514.  doi: 10.1016/j.jde.2005.07.019.

[37]

J. Wirth, Wave equations with time dependent dissipation. Ⅱ. Effective dissipation, J. Diff. Equa., 232 (2007), 74-103.  doi: 10.1016/j.jde.2006.06.004.

[38]

B. Yordanov and Q. S. Zhang, Finite time blow up for critical wave equations in high dimensions, J. Func. Anal., 231 (2006), 361-374.  doi: 10.1016/j.jfa.2005.03.012.

[39]

Q. S. Zhang, A blow-up result for a nonlinear wave equation with damping: The critical case, Comp. Rend. Acad. Scie. Pari. Séri. I Math., 333 (2001), 109-114.  doi: 10.1016/S0764-4442(01)01999-1.

[40]

Y. Zhou, Cauchy problem for semilinear wave equations in four space dimensions with small initial data, J. Part. Diff. Equa., 8 (1995), 135-144. 

[41]

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

[42]

Y. Zhou, Blow up of solutions to semilinear wave equations with critical exponent in high dimensions, Chin. Annu. Math. Seri. B, 28 (2007), 205-212.  doi: 10.1007/s11401-005-0205-x.

[43]

Y. Zhou and W. Han, Lifespan of solutions to critical semilinear wave equations, Comm. Part. Diff. Equa., 39 (2014), 439-451.  doi: 10.1080/03605302.2013.863914.

Figure 1.  wave equation with power nonlinearity
Figure 2.  wave equation with derivative nonlinearity
Figure 3.  wave equation with combined nonlinearities
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