A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain

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December 2020, 19(12): 5387-5411. doi: 10.3934/cpaa.2020243

A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain

Ahmad Z. Fino ^1,, and Wenhui Chen ^2,

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

^* Corresponding author

Received April 2020 Revised July 2020 Published December 2020 Early access September 2020

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.

Keywords: Semilinear damped wave equation, exterior domain, strong damping, mixed nonlinearity, global existence.

Mathematics Subject Classification: Primary: 35A01; Secondary: 35L20, 35L05.

Citation: Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure and Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

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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