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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 & Continuous Dynamical Systems - S, 2021, 14 (12) : 4575-4608. doi: 10.3934/dcdss.2021130
References:
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show all references

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

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

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

[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.  Google Scholar

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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.  Google Scholar

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[22]

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[32]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

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

[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.  Google Scholar

[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.  Google Scholar

[41]

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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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