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A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain
1. | Department of Mathematics, Faculty of Sciences, Lebanese University, Tripoli, P.O. Box 1352, Lebanon |
2. | Faculty of Mathematics and Computer Science, Technical University Bergakademie Freiberg, Freiberg, 09596, Germany |
We study two-dimensional semilinear strongly damped wave equation with mixed nonlinearity $ |u|^p+|u_t|^q $ in an exterior domain, where $ p, q>1 $. We prove global (in time) existence of small data solution with suitable higher regularity by using a weighted energy method, and assuming some conditions on powers of nonlinearity.
References:
[1] |
W. Chen and A. Z. Fino, Blow-up of solutions to semilinear strongly damped wave equations with different nonlinear terms in an exterior domain, arXiv: 1910.05981. |
[2] |
W. Chen and M. Reissig,
Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D, Math. Methods Appl. Sci., 42 (2019), 667-709.
doi: 10.1002/mma.5370. |
[3] |
F. Crispo and P. Maremonti,
An interpolation inequality in exterior domains, Rend. Sem. Mat. Univ. Padova, 112 (2004), 11-39.
|
[4] |
S. Cui,
Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal., 43 (2001), 293-323.
doi: 10.1016/S0362-546X(99)00195-9. |
[5] |
M. D'Abbicco, H. Takeda and R. Ikehata, Critical exponent for semi-linear wave equations with double damping terms in exterior domains, NoDEA Nonlinear Differ. Equ. Appl., 26 (2019), 56.
doi: 10.1007/s00030-019-0603-5. |
[6] |
M. D'Abbicco and M. Reissig,
Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592.
doi: 10.1002/mma.2913. |
[7] |
L. D'Ambrosio and S. Lucente,
Nonlinear Liouville theorems for Grushin and Tricomi operators, J. Differ. Equ., 193 (2003), 511-541.
doi: 10.1016/S0022-0396(03)00138-4. |
[8] |
A. Z. Fino, Finite time blow up for wave equations with strong damping in an exterior domain, preprint, arXiv: 2695271. |
[9] |
A. Z. Fino, H. Ibrahim and A. Wehbe,
blow-up result for a nonlinear damped wave equation in exterior domain: the critical case, Comput. Math. Appl., 73 (2017), 2415-2420.
doi: 10.1016/j.camwa.2017.03.030. |
[10] |
N. Hayashi, E. I. Kaikina and P. I. Naumkin,
Damped wave equation with a critical nonlinearity on a half line, J. Anal. Appl., 2 (2004), 95-112.
|
[11] |
R. Ikehata,
Asymptotic profiles for wave equations with strong damping, J. Differ. Equ., 257 (2014), 2159-2177.
doi: 10.1016/j.jde.2014.05.031. |
[12] |
R. Ikehata,
Global existence of solutions for semilinear damped wave equation in 2-D exterior domain, J. Differ. Equ., 200 (2004), 53-68.
doi: 10.1016/j.jde.2003.08.009. |
[13] |
R. Ikehata,
Critical exponent for semilinear damped wave equations in the N-dimensional half space, J. Math. Anal. Appl., 288 (2003), 803-818.
doi: 10.1016/j.jmaa.2003.09.029. |
[14] |
R. Ikehata,
Decay estimates of solutions for the wave equations with strong damping terms in unbounded domains, Math. Methods Appl. Sci., 24 (2001), 659-670.
doi: 10.1002/mma.235. |
[15] |
R. Ikehata and Y. Inoue,
Global existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain, Nonlinear Anal., 68 (2008), 154-169.
doi: 10.1016/j.na.2006.10.038. |
[16] |
R. Ikehata and K. Tanizawa,
Global existence of solutions for semilinear damped wave equations in $\mathbf{R}^N$ with non compactly supported initial data, Nonlinear Anal., 61 (2005), 1189-1208.
doi: 10.1016/j.na.2005.01.097. |
[17] |
R. Ikehata, G. Todorova and B. Yordanov,
Wave equations with strong damping in Hilbert spaces, J. Differ. Equ., 254 (2013), 3352-3368.
doi: 10.1016/j.jde.2013.01.023. |
[18] |
N. Lai and S. Yin,
Finite time blow-up for a kind of initial-boundary value problem of semilinear damped wave equation, Math. Methods Appl. Sci., 40 (2017), 1223-1230.
doi: 10.1002/mma.4046. |
[19] |
A. Mohammed Djouti and M. Reissig,
Weakly coupled systems of semilinear effectively damped waves with time-dependent coefficient, different power nonlinearities and different regularity of the data, Nonlinear Anal., 175 (2018), 28-55.
doi: 10.1016/j.na.2018.05.006. |
[20] |
K. Ono,
Decay estimates for dissipative wave equations in exterior domains, J. Math. Anal. Appl., 286 (2003), 540-562.
doi: 10.1016/S0022-247X(03)00489-X. |
[21] |
T. Ogawa and H. Takeda,
Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal., 70 (2009), 3696-3701.
doi: 10.1016/j.na.2008.07.025. |
[22] |
A. Palmieri and M. Reissig,
Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, II, Math. Nachr., 291 (2018), 1859-1892.
doi: 10.1002/mana.201700144. |
[23] |
G. Ponce,
Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418.
doi: 10.1016/0362-546X(85)90001-X. |
[24] |
Y. Shibata,
On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci., 23 (2000), 203-226.
doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M. |
[25] |
M. Sobajima,
Global existence of solutions to semilinear damped wave equation with slowly decaying initial data in exterior domain, Differ. Integral Equ., 32 (2019), 615-638.
|
[26] |
M. Sobajima and Y. Wakasugi, Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data, Commun. Contemp. Math., 21 (2019), 30 pp.
doi: 10.1142/S0219199718500359. |
[27] |
G. Todorova and B. Yordanov,
Critical exponent for a nonlinear wave equation with damping, J. Differ. Equ., 174 (2001), 464-489.
doi: 10.1006/jdeq.2000.3933. |
[28] |
Y. Wakasugi, On the diffusive structure for the damped wave equation with variable coefficients, Ph.D thesis, Osaka University, 2014. |
show all references
References:
[1] |
W. Chen and A. Z. Fino, Blow-up of solutions to semilinear strongly damped wave equations with different nonlinear terms in an exterior domain, arXiv: 1910.05981. |
[2] |
W. Chen and M. Reissig,
Weakly coupled systems of semilinear elastic waves with different damping mechanisms in 3D, Math. Methods Appl. Sci., 42 (2019), 667-709.
doi: 10.1002/mma.5370. |
[3] |
F. Crispo and P. Maremonti,
An interpolation inequality in exterior domains, Rend. Sem. Mat. Univ. Padova, 112 (2004), 11-39.
|
[4] |
S. Cui,
Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal., 43 (2001), 293-323.
doi: 10.1016/S0362-546X(99)00195-9. |
[5] |
M. D'Abbicco, H. Takeda and R. Ikehata, Critical exponent for semi-linear wave equations with double damping terms in exterior domains, NoDEA Nonlinear Differ. Equ. Appl., 26 (2019), 56.
doi: 10.1007/s00030-019-0603-5. |
[6] |
M. D'Abbicco and M. Reissig,
Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592.
doi: 10.1002/mma.2913. |
[7] |
L. D'Ambrosio and S. Lucente,
Nonlinear Liouville theorems for Grushin and Tricomi operators, J. Differ. Equ., 193 (2003), 511-541.
doi: 10.1016/S0022-0396(03)00138-4. |
[8] |
A. Z. Fino, Finite time blow up for wave equations with strong damping in an exterior domain, preprint, arXiv: 2695271. |
[9] |
A. Z. Fino, H. Ibrahim and A. Wehbe,
blow-up result for a nonlinear damped wave equation in exterior domain: the critical case, Comput. Math. Appl., 73 (2017), 2415-2420.
doi: 10.1016/j.camwa.2017.03.030. |
[10] |
N. Hayashi, E. I. Kaikina and P. I. Naumkin,
Damped wave equation with a critical nonlinearity on a half line, J. Anal. Appl., 2 (2004), 95-112.
|
[11] |
R. Ikehata,
Asymptotic profiles for wave equations with strong damping, J. Differ. Equ., 257 (2014), 2159-2177.
doi: 10.1016/j.jde.2014.05.031. |
[12] |
R. Ikehata,
Global existence of solutions for semilinear damped wave equation in 2-D exterior domain, J. Differ. Equ., 200 (2004), 53-68.
doi: 10.1016/j.jde.2003.08.009. |
[13] |
R. Ikehata,
Critical exponent for semilinear damped wave equations in the N-dimensional half space, J. Math. Anal. Appl., 288 (2003), 803-818.
doi: 10.1016/j.jmaa.2003.09.029. |
[14] |
R. Ikehata,
Decay estimates of solutions for the wave equations with strong damping terms in unbounded domains, Math. Methods Appl. Sci., 24 (2001), 659-670.
doi: 10.1002/mma.235. |
[15] |
R. Ikehata and Y. Inoue,
Global existence of weak solutions for two-dimensional semilinear wave equations with strong damping in an exterior domain, Nonlinear Anal., 68 (2008), 154-169.
doi: 10.1016/j.na.2006.10.038. |
[16] |
R. Ikehata and K. Tanizawa,
Global existence of solutions for semilinear damped wave equations in $\mathbf{R}^N$ with non compactly supported initial data, Nonlinear Anal., 61 (2005), 1189-1208.
doi: 10.1016/j.na.2005.01.097. |
[17] |
R. Ikehata, G. Todorova and B. Yordanov,
Wave equations with strong damping in Hilbert spaces, J. Differ. Equ., 254 (2013), 3352-3368.
doi: 10.1016/j.jde.2013.01.023. |
[18] |
N. Lai and S. Yin,
Finite time blow-up for a kind of initial-boundary value problem of semilinear damped wave equation, Math. Methods Appl. Sci., 40 (2017), 1223-1230.
doi: 10.1002/mma.4046. |
[19] |
A. Mohammed Djouti and M. Reissig,
Weakly coupled systems of semilinear effectively damped waves with time-dependent coefficient, different power nonlinearities and different regularity of the data, Nonlinear Anal., 175 (2018), 28-55.
doi: 10.1016/j.na.2018.05.006. |
[20] |
K. Ono,
Decay estimates for dissipative wave equations in exterior domains, J. Math. Anal. Appl., 286 (2003), 540-562.
doi: 10.1016/S0022-247X(03)00489-X. |
[21] |
T. Ogawa and H. Takeda,
Non-existence of weak solutions to nonlinear damped wave equations in exterior domains, Nonlinear Anal., 70 (2009), 3696-3701.
doi: 10.1016/j.na.2008.07.025. |
[22] |
A. Palmieri and M. Reissig,
Semi-linear wave models with power non-linearity and scale-invariant time-dependent mass and dissipation, II, Math. Nachr., 291 (2018), 1859-1892.
doi: 10.1002/mana.201700144. |
[23] |
G. Ponce,
Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418.
doi: 10.1016/0362-546X(85)90001-X. |
[24] |
Y. Shibata,
On the rate of decay of solutions to linear viscoelastic equation, Math. Methods Appl. Sci., 23 (2000), 203-226.
doi: 10.1002/(SICI)1099-1476(200002)23:3<203::AID-MMA111>3.0.CO;2-M. |
[25] |
M. Sobajima,
Global existence of solutions to semilinear damped wave equation with slowly decaying initial data in exterior domain, Differ. Integral Equ., 32 (2019), 615-638.
|
[26] |
M. Sobajima and Y. Wakasugi, Weighted energy estimates for wave equation with space-dependent damping term for slowly decaying initial data, Commun. Contemp. Math., 21 (2019), 30 pp.
doi: 10.1142/S0219199718500359. |
[27] |
G. Todorova and B. Yordanov,
Critical exponent for a nonlinear wave equation with damping, J. Differ. Equ., 174 (2001), 464-489.
doi: 10.1006/jdeq.2000.3933. |
[28] |
Y. Wakasugi, On the diffusive structure for the damped wave equation with variable coefficients, Ph.D thesis, Osaka University, 2014. |
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