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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

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  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.

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 & 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. Google Scholar

[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.  Google Scholar

[3]

F. Crispo and P. Maremonti, An interpolation inequality in exterior domains, Rend. Sem. Mat. Univ. Padova, 112 (2004), 11-39.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913.  Google Scholar

[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.  Google Scholar

[8]

A. Z. Fino, Finite time blow up for wave equations with strong damping in an exterior domain, preprint, arXiv: 2695271. Google Scholar

[9]

A. Z. FinoH. 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.  Google Scholar

[10]

N. HayashiE. I. Kaikina and P. I. Naumkin, Damped wave equation with a critical nonlinearity on a half line, J. Anal. Appl., 2 (2004), 95-112.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

R. IkehataG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

Y. Wakasugi, On the diffusive structure for the damped wave equation with variable coefficients, Ph.D thesis, Osaka University, 2014. Google Scholar

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. Google Scholar

[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.  Google Scholar

[3]

F. Crispo and P. Maremonti, An interpolation inequality in exterior domains, Rend. Sem. Mat. Univ. Padova, 112 (2004), 11-39.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

M. D'Abbicco and M. Reissig, Semilinear structural damped waves, Math. Methods Appl. Sci., 37 (2014), 1570-1592.  doi: 10.1002/mma.2913.  Google Scholar

[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.  Google Scholar

[8]

A. Z. Fino, Finite time blow up for wave equations with strong damping in an exterior domain, preprint, arXiv: 2695271. Google Scholar

[9]

A. Z. FinoH. 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.  Google Scholar

[10]

N. HayashiE. I. Kaikina and P. I. Naumkin, Damped wave equation with a critical nonlinearity on a half line, J. Anal. Appl., 2 (2004), 95-112.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

R. IkehataG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[28]

Y. Wakasugi, On the diffusive structure for the damped wave equation with variable coefficients, Ph.D thesis, Osaka University, 2014. Google Scholar

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