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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:
 [1] 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. Andrews, K. 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 [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.  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 [6] G. Autuori, P. 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 [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.  Google Scholar [8] 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 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] L. Bociu, M. 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.  Google Scholar [14] H. Brezis, Functional Analysis, SObolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar [15] M. M. Cavalcanti, V. 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.  Google Scholar [16] M. M. Cavalcanti, V. 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.  Google Scholar [17] I. Chueshov, M. 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.  Google Scholar [18] P. G. Ciarlet, Linear and Nonlinear Functional Analysis with Applications, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2013.  Google Scholar [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.  Google Scholar [20] Darmawijoyo and W. 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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. Gal, G. 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. 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##### References:
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Anal., 196 (2010), 489-516.  doi: 10.1007/s00205-009-0241-x.  Google Scholar [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.  Google Scholar [8] 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 [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] L. Bociu, M. 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.  Google Scholar [14] H. Brezis, Functional Analysis, SObolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar [15] M. M. Cavalcanti, V. N. Domingos Cavalcanti and I. Lasiecka, Well-posedness and optimal decay rates for the wave equation with nonlinear boundary damping–-source interaction, J. 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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. Kashiwabara, C. M. Colciago, L. 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. 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The regions covered by (6), and their subregions in which (14) holds or not, in dimensions $N = 2,3,4$
(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) ${\mathcal {X}}$ $q>2$ $p>2$ $p>2$, $q>2$
(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) $p>2$ $p>2$, $\overline{\mu}<1+p/2$ $p>\overline{m}$ $p>\overline{m}$, $\overline{\mu}<1+p/2$
(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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