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December  2021, 14(12): 4575-4608. doi: 10.3934/dcdss.2021130

Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources

Dipartimento di Matematica ed Informatica, Università di Perugia, Via Vanvitelli, 1 06123 Perugia, Italy

Received  July 2021 Published  December 2021 Early access  October 2021

Fund Project: The work was realized within the auspices of the INdAM – GNAMPA Projects Equazioni alle derivate parziali: Problemi e Modelli (Prot_U-UFMBAZ-2020-000761), and it was also supported by Progetto Equazione delle onde con condizioni acustiche, finanziato con il Fondo Ricerca di Base, 2019, della Università degli Studi di Perugia and by Progetti Equazioni delle onde con condizioni iperboliche ed acustiche al bordo, finanziati con i Fondi Ricerca di Base 2017 and 2018, della Università degli Studi di Perugia

The aim of this paper is to give global nonexistence and blow–up results for the problem
$ \begin{cases} u_{tt}-\Delta u+P(x,u_t) = f(x,u) \qquad &\text{in $(0,\infty)\times\Omega$,}\\ u = 0 &\text{on $(0,\infty)\times \Gamma_0$,}\\ u_{tt}+\partial_\nu u-\Delta_\Gamma u+Q(x,u_t) = g(x,u)\qquad &\text{on $(0,\infty)\times \Gamma_1$,}\\ u(0,x) = u_0(x),\quad u_t(0,x) = u_1(x) & \text{in $\overline{\Omega}$,} \end{cases} $
where
$ \Omega $
is a bounded open
$ C^1 $
subset of
$ {\mathbb R}^N $
,
$ N\ge 2 $
,
$ \Gamma = \partial\Omega $
,
$ (\Gamma_0,\Gamma_1) $
is a partition of
$ \Gamma $
,
$ \Gamma_1\not = \emptyset $
being relatively open in
$ \Gamma $
,
$ \Delta_\Gamma $
denotes the Laplace–Beltrami operator on
$ \Gamma $
,
$ \nu $
is the outward normal to
$ \Omega $
, and the terms
$ P $
and
$ Q $
represent nonlinear damping terms, while
$ f $
and
$ g $
are nonlinear source terms. These results complement the analysis of the problem given by the author in two recent papers, dealing with local and global existence, uniqueness and well–posedness.
Citation: Enzo Vitillaro. Blow–up for the wave equation with hyperbolic dynamical boundary conditions, interior and boundary nonlinear damping and sources. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4575-4608. doi: 10.3934/dcdss.2021130
References:
[1]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[2] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge University Press, Cambridge, 1993. 
[3]

K. T. AndrewsK. L. Kuttler and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795.  doi: 10.1006/jmaa.1996.0053.

[4]

G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal., 73 (2010), 1952-1965.  doi: 10.1016/j.na.2010.05.024.

[5]

G. Autuori and P. Pucci, Kirchhoff systems with nonlinear source and boundary damping terms, Commun. Pure Appl. Anal., 9 (2010), 1161-1188.  doi: 10.3934/cpaa.2010.9.1161.

[6]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.

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J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917.  doi: 10.1512/iumj.1976.25.25071.

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J. Bergh and J. Lőfstrőm, Interpolations Spaces. An Introduction, Springer Verlag, Berlin, 1976.

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L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping, Nonlinear Anal., 71 (2009), e560–e575. doi: 10.1016/j.na.2008.11.062.

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L. Bociu and I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Appl. Math. (Warsaw), 35 (2008), 281-304.  doi: 10.4064/am35-3-3.

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L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860.  doi: 10.3934/dcds.2008.22.835.

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L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.  doi: 10.1016/j.jde.2010.03.009.

[13]

L. BociuM. Rammaha and D. Toundykov, On a wave equation with supercritical interior and boundary sources and damping terms, Math. Nachr., 284 (2011), 2032-2064.  doi: 10.1002/mana.200910182.

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H. Brezis, Functional Analysis, SObolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

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M. M. CavalcantiV. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–-source interaction, J. Differential Equations, 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.

[16]

M. M. CavalcantiV. N. Domingos Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158.  doi: 10.1016/j.jde.2004.04.011.

[17]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.

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Darmawijoyo and W. T. van Horssen, On boundary damping for a weakly nonlinear wave equation, Nonlinear Dynam., 30 (2002), 179-191.  doi: 10.1023/A:1020473930223.

[21]

G. G. Doronin, N. A. Lar'kin and A. J. Souza, A hyperbolic problem with nonlinear second-order boundary damping, Electron. J. Differential Equations, (1998), 10pp. (electronic).

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A. FaviniC. G. GalJ. A. GoldsteinG. R. Goldstein and S. Romanelli, The non-autonomous wave equation with general Wentzell boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 317-329.  doi: 10.1017/S0308210500003905.

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H. FengS. Li and X. Zhi, Blow-up solutions for a nonlinear wave equation with boundary damping and interior source, Nonlinear Anal., 75 (2012), 2273-2280.  doi: 10.1016/j.na.2011.10.027.

[24]

A. Figotin and G. Reyes, Lagrangian variational framework for boundary value problems, J. Math. Phys., 56 (2015), 1-35.  doi: 10.1063/1.4931135.

[25]

A. Fiscella and E. Vitillaro, Blow-up for the wave equation with nonlinear source and boundary damping terms, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 759-778.  doi: 10.1017/S0308210515000165.

[26]

N. Fourrier and I. Lasiecka, Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Equ. Control Theory, 2 (2013), 631-667.  doi: 10.3934/eect.2013.2.631.

[27]

C. G. GalG. R. Goldstein and J. A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, J. Evol. Equ., 3 (2003), 623-635.  doi: 10.1007/s00028-003-0113-z.

[28]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.

[29]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. 

[30]

P. J. Graber, Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping, J. Evol. Equ., 12 (2012), 141-164.  doi: 10.1007/s00028-011-0127-x.

[31]

P. J. Graber and I. Lasiecka, Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions, Semigroup Forum, 88 (2014), 333-365.  doi: 10.1007/s00233-013-9534-3.

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B. Guo and C.-Z. Xu, On the spectrum-determined growth condition of a vibration cable with a tip mass, IEEE Trans. Automat. Control, 45 (2000), 89-93.  doi: 10.1109/9.827360.

[34]

T. G. Ha, Blow-up for semilinear wave equation with boundary damping and source terms, J. Math. Anal. Appl., 390 (2012), 328-334.  doi: 10.1016/j.jmaa.2012.01.037.

[35]

T. G. Ha, Blow-up for wave equation with weak boundary damping and source terms, Appl. Math. Lett., 49 (2015), 166-172.  doi: 10.1016/j.aml.2015.05.003.

[36]

T. KashiwabaraC. M. ColciagoL. Dedè and and A. Quarteroni, Well-posedness, regularity, and convergence analysis of the finite element approximation of a generalized Robin boundary value problem, SIAM J. Numer. Anal., 53 (2015), 105-126.  doi: 10.1137/140954477.

[37]

I. Lasiecka and L. Bociu, Well-posedness and blow-up of solutions to wave equations with supercritical boundary sources and boundary damping, In Differential & Difference Equations and Applications, Hindawi Publ. Corp., New York, (2006), 635–643.

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I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equation, 6 (1993), 507-533. 

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H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361.  doi: 10.1007/s002050050032.

[40]

W. Littman and B. Liu, On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math., 59 (1999), 17–34 (electronic). doi: 10.1137/S0036139996314106.

[41]

W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Arch. Rational Mech. Anal., 103 (1988), 193-236.  doi: 10.1007/BF00251758.

[42]

W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura Appl. (4), 152 (1988), 281-330.  doi: 10.1007/BF01766154.

[43]

T. Meurer and A. Kugi, Tracking control design for a wave equation with dynamic boundary conditions modeling a piezoelectric stack actuator, Internat. J. Robust Nonlinear Control, 21 (2011), 542-562.  doi: 10.1002/rnc.1611.

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Ö. MorgülB. Rao and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Trans. Automat. Control, 39 (1994), 2140-2145.  doi: 10.1109/9.328811.

[45] P. Morse and K. Ingard, Theoretical Acoustics, International series in pure and applied physics, Princeton University Press, 1968. 
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D. Mugnolo, Abstract wave equations with acoustic boundary conditions, Math. Nachr., 279 (2006), 299-318.  doi: 10.1002/mana.200310362.

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D. Mugnolo and E. Vitillaro, The wave equation with acoustic boundary conditions on non-locally reacting surfaces, 2021, arXiv: 2105.09219

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P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214.  doi: 10.1006/jdeq.1998.3477.

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P. Radu, Weak solutions to the Cauchy problem of a semilinear wave equation with damping and source terms, Adv. Differential Equations, 10 (2005), 1261-1300. 

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show all references

References:
[1]

R. A. Adams, Sobolev Spaces, Pure and Applied Mathematics, Vol. 65. Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.

[2] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge University Press, Cambridge, 1993. 
[3]

K. T. AndrewsK. L. Kuttler and M. Shillor, Second order evolution equations with dynamic boundary conditions, J. Math. Anal. Appl., 197 (1996), 781-795.  doi: 10.1006/jmaa.1996.0053.

[4]

G. Autuori and P. Pucci, Kirchhoff systems with dynamic boundary conditions, Nonlinear Anal., 73 (2010), 1952-1965.  doi: 10.1016/j.na.2010.05.024.

[5]

G. Autuori and P. Pucci, Kirchhoff systems with nonlinear source and boundary damping terms, Commun. Pure Appl. Anal., 9 (2010), 1161-1188.  doi: 10.3934/cpaa.2010.9.1161.

[6]

G. AutuoriP. Pucci and M. C. Salvatori, Global nonexistence for nonlinear Kirchhoff systems, Arch. Ration. Mech. Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.

[7]

J. T. Beale, Spectral properties of an acoustic boundary condition, Indiana Univ. Math. J., 25 (1976), 895-917.  doi: 10.1512/iumj.1976.25.25071.

[8]

J. Bergh and J. Lőfstrőm, Interpolations Spaces. An Introduction, Springer Verlag, Berlin, 1976.

[9]

L. Bociu, Local and global wellposedness of weak solutions for the wave equation with nonlinear boundary and interior sources of supercritical exponents and damping, Nonlinear Anal., 71 (2009), e560–e575. doi: 10.1016/j.na.2008.11.062.

[10]

L. Bociu and I. Lasiecka, Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping, Appl. Math. (Warsaw), 35 (2008), 281-304.  doi: 10.4064/am35-3-3.

[11]

L. Bociu and I. Lasiecka, Uniqueness of weak solutions for the semilinear wave equations with supercritical boundary/interior sources and damping, Discrete Contin. Dyn. Syst., 22 (2008), 835-860.  doi: 10.3934/dcds.2008.22.835.

[12]

L. Bociu and I. Lasiecka, Local Hadamard well-posedness for nonlinear wave equations with supercritical sources and damping, J. Differential Equations, 249 (2010), 654-683.  doi: 10.1016/j.jde.2010.03.009.

[13]

L. BociuM. Rammaha and D. Toundykov, On a wave equation with supercritical interior and boundary sources and damping terms, Math. Nachr., 284 (2011), 2032-2064.  doi: 10.1002/mana.200910182.

[14]

H. Brezis, Functional Analysis, SObolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[15]

M. M. CavalcantiV. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–-source interaction, J. Differential Equations, 236 (2007), 407-459.  doi: 10.1016/j.jde.2007.02.004.

[16]

M. M. CavalcantiV. N. Domingos Cavalcanti and P. Martinez, Existence and decay rate estimates for the wave equation with nonlinear boundary damping and source term, J. Differential Equations, 203 (2004), 119-158.  doi: 10.1016/j.jde.2004.04.011.

[17]

I. ChueshovM. Eller and I. Lasiecka, On the attractor for a semilinear wave equation with critical exponent and nonlinear boundary dissipation, Comm. Partial Differential Equations, 27 (2002), 1901-1951.  doi: 10.1081/PDE-120016132.

[18]

P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013.

[19]

F. Conrad and Ö. Morgül, On the stabilization of a flexible beam with a tip mass, SIAM J. Control Optim., 36 (1998), 1962-1986.  doi: 10.1137/S0363012996302366.

[20]

Darmawijoyo and W. T. van Horssen, On boundary damping for a weakly nonlinear wave equation, Nonlinear Dynam., 30 (2002), 179-191.  doi: 10.1023/A:1020473930223.

[21]

G. G. Doronin, N. A. Lar'kin and A. J. Souza, A hyperbolic problem with nonlinear second-order boundary damping, Electron. J. Differential Equations, (1998), 10pp. (electronic).

[22]

A. FaviniC. G. GalJ. A. GoldsteinG. R. Goldstein and S. Romanelli, The non-autonomous wave equation with general Wentzell boundary conditions, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 317-329.  doi: 10.1017/S0308210500003905.

[23]

H. FengS. Li and X. Zhi, Blow-up solutions for a nonlinear wave equation with boundary damping and interior source, Nonlinear Anal., 75 (2012), 2273-2280.  doi: 10.1016/j.na.2011.10.027.

[24]

A. Figotin and G. Reyes, Lagrangian variational framework for boundary value problems, J. Math. Phys., 56 (2015), 1-35.  doi: 10.1063/1.4931135.

[25]

A. Fiscella and E. Vitillaro, Blow-up for the wave equation with nonlinear source and boundary damping terms, Proc. Roy. Soc. Edinburgh Sect. A, 145 (2015), 759-778.  doi: 10.1017/S0308210515000165.

[26]

N. Fourrier and I. Lasiecka, Regularity and stability of a wave equation with a strong damping and dynamic boundary conditions, Evol. Equ. Control Theory, 2 (2013), 631-667.  doi: 10.3934/eect.2013.2.631.

[27]

C. G. GalG. R. Goldstein and J. A. Goldstein, Oscillatory boundary conditions for acoustic wave equations, J. Evol. Equ., 3 (2003), 623-635.  doi: 10.1007/s00028-003-0113-z.

[28]

V. Georgiev and G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109 (1994), 295-308.  doi: 10.1006/jdeq.1994.1051.

[29]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. 

[30]

P. J. Graber, Uniform boundary stabilization of a wave equation with nonlinear acoustic boundary conditions and nonlinear boundary damping, J. Evol. Equ., 12 (2012), 141-164.  doi: 10.1007/s00028-011-0127-x.

[31]

P. J. Graber and I. Lasiecka, Analyticity and Gevrey class regularity for a strongly damped wave equation with hyperbolic dynamic boundary conditions, Semigroup Forum, 88 (2014), 333-365.  doi: 10.1007/s00233-013-9534-3.

[32]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985.

[33]

B. Guo and C.-Z. Xu, On the spectrum-determined growth condition of a vibration cable with a tip mass, IEEE Trans. Automat. Control, 45 (2000), 89-93.  doi: 10.1109/9.827360.

[34]

T. G. Ha, Blow-up for semilinear wave equation with boundary damping and source terms, J. Math. Anal. Appl., 390 (2012), 328-334.  doi: 10.1016/j.jmaa.2012.01.037.

[35]

T. G. Ha, Blow-up for wave equation with weak boundary damping and source terms, Appl. Math. Lett., 49 (2015), 166-172.  doi: 10.1016/j.aml.2015.05.003.

[36]

T. KashiwabaraC. M. ColciagoL. Dedè and and A. Quarteroni, Well-posedness, regularity, and convergence analysis of the finite element approximation of a generalized Robin boundary value problem, SIAM J. Numer. Anal., 53 (2015), 105-126.  doi: 10.1137/140954477.

[37]

I. Lasiecka and L. Bociu, Well-posedness and blow-up of solutions to wave equations with supercritical boundary sources and boundary damping, In Differential & Difference Equations and Applications, Hindawi Publ. Corp., New York, (2006), 635–643.

[38]

I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differential Integral Equation, 6 (1993), 507-533. 

[39]

H. A. Levine and J. Serrin, Global nonexistence theorems for quasilinear evolution equations with dissipation, Arch. Rational Mech. Anal., 137 (1997), 341-361.  doi: 10.1007/s002050050032.

[40]

W. Littman and B. Liu, On the spectral properties and stabilization of acoustic flow, SIAM J. Appl. Math., 59 (1999), 17–34 (electronic). doi: 10.1137/S0036139996314106.

[41]

W. Littman and L. Markus, Exact boundary controllability of a hybrid system of elasticity, Arch. Rational Mech. Anal., 103 (1988), 193-236.  doi: 10.1007/BF00251758.

[42]

W. Littman and L. Markus, Stabilization of a hybrid system of elasticity by feedback boundary damping, Ann. Mat. Pura Appl. (4), 152 (1988), 281-330.  doi: 10.1007/BF01766154.

[43]

T. Meurer and A. Kugi, Tracking control design for a wave equation with dynamic boundary conditions modeling a piezoelectric stack actuator, Internat. J. Robust Nonlinear Control, 21 (2011), 542-562.  doi: 10.1002/rnc.1611.

[44]

Ö. MorgülB. Rao and F. Conrad, On the stabilization of a cable with a tip mass, IEEE Trans. Automat. Control, 39 (1994), 2140-2145.  doi: 10.1109/9.328811.

[45] P. Morse and K. Ingard, Theoretical Acoustics, International series in pure and applied physics, Princeton University Press, 1968. 
[46]

D. Mugnolo, Abstract wave equations with acoustic boundary conditions, Math. Nachr., 279 (2006), 299-318.  doi: 10.1002/mana.200310362.

[47]

D. Mugnolo and E. Vitillaro, The wave equation with acoustic boundary conditions on non-locally reacting surfaces, 2021, arXiv: 2105.09219

[48]

P. Pucci and J. Serrin, Global nonexistence for abstract evolution equations with positive initial energy, J. Differential Equations, 150 (1998), 203-214.  doi: 10.1006/jdeq.1998.3477.

[49]

P. Radu, Weak solutions to the Cauchy problem of a semilinear wave equation with damping and source terms, Adv. Differential Equations, 10 (2005), 1261-1300. 

[50]

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Figure 1.  The regions covered by (6), and their subregions in which (14) holds or not, in dimensions $ N = 2,3,4 $
Table 1.  (18) vs. (19) for the model classes A–D)
Case (16) A) $\gamma = 0$, $\delta = 0$ B) $\gamma = 0$, $\delta>0$ C) $\gamma>0$, $\delta = 0$ D) $\gamma>0$, $\delta>0$
(18) $\checkmark$ $q = 2$ $p = 2$ $p = 2$, $q = 2$
(19) ${\mathcal {X}}$ $q>2$ $p>2$ $p>2$, $q>2$
Case (16) A) $\gamma = 0$, $\delta = 0$ B) $\gamma = 0$, $\delta>0$ C) $\gamma>0$, $\delta = 0$ D) $\gamma>0$, $\delta>0$
(18) $\checkmark$ $q = 2$ $p = 2$ $p = 2$, $q = 2$
(19) ${\mathcal {X}}$ $q>2$ $p>2$ $p>2$, $q>2$
Table 2.  (14) vs. (22) for the model problems Ca–Cd)
$\gamma>0 = \delta$ $\alpha = \beta = 0$ $\alpha = 0<\beta$ $\alpha>0 = \beta$ $\alpha,\beta>0$
(14) $p = 2$ $p = 2$ $p\le\overline{m}$ $p\le\overline{m}$
(22) $p>2$ $p>2$, $\overline{\mu}<1+p/2$ $p>\overline{m}$ $p>\overline{m}$, $\overline{\mu}<1+p/2$
$\gamma>0 = \delta$ $\alpha = \beta = 0$ $\alpha = 0<\beta$ $\alpha>0 = \beta$ $\alpha,\beta>0$
(14) $p = 2$ $p = 2$ $p\le\overline{m}$ $p\le\overline{m}$
(22) $p>2$ $p>2$, $\overline{\mu}<1+p/2$ $p>\overline{m}$ $p>\overline{m}$, $\overline{\mu}<1+p/2$
Table 3.  (14) vs (26) for the model problems Da–Dd)
$\gamma,\delta>0$ $\alpha = \beta = 0$ $\alpha = 0<\beta$ $\alpha>0 = \beta$ $\alpha,\beta>0$
(14) $p = 2$,$q = 2$ $p = 2$, $q\le \overline{\mu}$ $p\le\overline{m}$, $q = 2$ $p\le\overline{m}$, $q\le\overline{\mu}$
(26) $p>2$, $q>2$ $p>2$, $q> \begin{cases}2 & \text { if } \bar{\mu} <1+\frac{p}{2} \\ \bar{\mu} & \text { if } \bar{\mu} \geq 1+\frac{p}{2}\end{cases}$ $p>\overline{m}$, q>2 $p>\overline{m}$, $q> \begin{cases}2 & \text { if } \bar{\mu} <1+\frac{p}{2} \\ \bar{\mu} & \text { if } \bar{\mu} \geq 1+\frac{p}{2}\end{cases}$
$\gamma,\delta>0$ $\alpha = \beta = 0$ $\alpha = 0<\beta$ $\alpha>0 = \beta$ $\alpha,\beta>0$
(14) $p = 2$,$q = 2$ $p = 2$, $q\le \overline{\mu}$ $p\le\overline{m}$, $q = 2$ $p\le\overline{m}$, $q\le\overline{\mu}$
(26) $p>2$, $q>2$ $p>2$, $q> \begin{cases}2 & \text { if } \bar{\mu} <1+\frac{p}{2} \\ \bar{\mu} & \text { if } \bar{\mu} \geq 1+\frac{p}{2}\end{cases}$ $p>\overline{m}$, q>2 $p>\overline{m}$, $q> \begin{cases}2 & \text { if } \bar{\mu} <1+\frac{p}{2} \\ \bar{\mu} & \text { if } \bar{\mu} \geq 1+\frac{p}{2}\end{cases}$
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