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Blow-up in damped abstract nonlinear equations

  • Received: 01 December 2019 Revised: 01 February 2020
  • Primary: 35L70, 35L90, 35B44, 35L05; Secondary: 35B35, 35B40

  • As a typical example of our analysis we consider a generalized Boussinesq equation, linearly damped and with a nonlinear source term. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We do our analysis in an abstract framework. We compare our results with those in the literature and we give more examples to illustrate the applicability of the abstract formulation.

    Citation: Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations[J]. Electronic Research Archive, 2020, 28(1): 347-367. doi: 10.3934/era.2020020

    Related Papers:

  • As a typical example of our analysis we consider a generalized Boussinesq equation, linearly damped and with a nonlinear source term. For any positive value of the initial energy, in particular for high energies, we give sufficient conditions on the initial data to conclude nonexistence of global solutions. We do our analysis in an abstract framework. We compare our results with those in the literature and we give more examples to illustrate the applicability of the abstract formulation.



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    [1] B. A. Bilgin and V. K. Kalantarov, Non-existence of global solutions to nonlinear wave equations with positive initial energy, Commun. Pure Appl. Anal., 17 (2018), no. 3,987–999. doi: 10.3934/cpaa.2018048
    [2] Théorie des ondes et des remous qui se propagent le long d'un canal rectangulaire en communiquant au liquide contenu dans ce canal des vitesses sensiblement pareilles de la surface au fond. J. Math. Pures Appl. (1872) 17: 55-108.
    [3] On Boussinesq's paradigm in nonlinear wave propagation. Comptes Rendus Mécanique (2007) 335: 521-535.
    [4] M. Dimova, N. Kolkovska and N. Kutev, Blow up of solutions to ordinary differential equations arising in nonlinear dispersive problems, Electron. J. Differential Equations, (2018) Paper No. 68, 16 pp.
    [5] M. Dimova, N. Kolkovska and N. Kutev, On the solvability of sixth order nonlinear double dispersive equations, AIP Conf. Proc., 2159 (2019), 030008, 10 pp. doi: 10.1063/1.5127473
    [6] R. Donninger and W. Schlag, Numerical study of the blowup/global existence dichotomy for the focusing cubic nonlinear Klein-Gordon equation, Nonlinearity, 24 (2011), no. 9, 2547–2562. doi: 10.1088/0951-7715/24/9/009
    [7] J. A. Esquivel-Avila, The dynamics of a nonlinear wave equation, J. Math. Anal. Appl., 279 (2003), no. 1,135–150. doi: 10.1016/S0022-247X(02)00701-1
    [8] J. A. Esquivel-Avila, Qualitative analysis of a nonlinear wave equation, Discrete Contin. Dyn. Syst., 10 (2004), no. 3,787–804. doi: 10.3934/dcds.2004.10.787
    [9] J. A. Esquivel-Avila, Dynamic analysis of a nonlinear Timoshenko equation, Abstr. Appl. Anal., 2011 (2011), Art. ID 724815, 36 pp. doi: 10.1155/2011/724815
    [10] J. A. Esquivel-Avila, Blow up and asymptotic behavior in a nondissipative nonlinear wave equation, Appl. Anal., 93 (2014), no. 9, 1963–1978. doi: 10.1080/00036811.2013.859250
    [11] J. A. Esquivel-Avila, Remarks on the qualitative behavior of the undamped Klein-Gordon equation, Math. Methods Appl. Sci., 41 (2018), no. 1,103–111. doi: 10.1002/mma.4598
    [12] J. A. Esquivel-Avila, Nonexistence of global solutions of abstract wave equations with high energies, J. Inequal. Appl., 2017 (2017) Paper No. 268, 14 pp. doi: 10.1186/s13660-017-1546-1
    [13] J. A. Esquivel-Avila, Global attractor for a nonlinear Timoshenko equation with source terms, Math. Sci. (Springer), 7 (2013), Art. 32, 8 pp. doi: 10.1186/2251-7456-7-32
    [14] Global solutions and finite time blow up for damped semilinear wave equations. Ann. Inst. H. Poincaré Anal. Non Linéaire (2006) 23: 185-207.
    [15] M. O. Korpusov, On blow-up of solutions to a Kirchhoff type dissipative wave equation with a source and positive energy, Sib. Math. J., 53 (2012), no. 4,702–717. doi: 10.1134/S003744661204012X
    [16] N. Kutev, N. Kolkovska and M. Dimova, Global behavior of the solutions to Boussinesq type equation with linear restoring force, AIP Conf. Proc., 1629 (2014), no. 1,172–185. doi: 10.1063/1.4902272
    [17] N. Kutev, N. Kolkovska, M. Dimova and C. I. Christov, Theoretical and numerical aspects for global existence and blow up for the solutions to Boussinesq Paradigm Equation, AIP Conf. Proc., 1404 (2011), no. 1, 68–76. doi: 10.1063/1.3659905
    [18] H. A. Levine and G. Todorova, Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy, Proc. Amer. Math. Soc., 129 (2001), no. 3,793–805. doi: 10.1090/S0002-9939-00-05743-9
    [19] W. Lian and R. Xu, Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term, Adv. Nonlinear Anal., 9 (2020), no. 1,613–632. doi: 10.1515/anona-2020-0016
    [20] W. Lian, R. Xu, V. D. Radulescu, Y. Yang and N. Zhao, Global well-posedness for a class of fourth order nonlinear strongly damped wave equations, Advances in Calculus of Variations, published online, (2019). doi: 10.1515/acv-2019-0039
    [21] Y. Liu and R. Xu, Potential well method for Cauchy problem of generalized double dispersion equations, J. Math. Anal. Appl., 338 (2008), no. 2, 1169–1187. doi: 10.1016/j.jmaa.2007.05.076
    [22] Y. Liu and R. Xu, Potential well method for initial boundary value problem of the generalized double dispersion equations, Commun. Pure Appl. Anal., 7 (2008), no. 1, 63–81. doi: 10.3934/cpaa.2008.7.63
    [23] (1999) Nonlinear Waves in Elastic Crystals.Oxford University Press.
    [24] L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), no. 3-4,273–303. doi: 10.1007/BF02761595
    [25] Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation. J. Math. Anal. Appl. (2009) 349: 10-20.
    [26] X. Su and S. Wang, The initial-boundary value problem for the generalized double dispersion equation, Z. Angew. Math. Phys., 68 (2017), Art. 53, 21 pp. doi: 10.1007/s00033-017-0798-4
    [27] G. Todorova and E. Vitillaro, Blow-up for nonlinear dissipative wave equations in $\mathbb{R}^n$, J. Math. Anal. Appl., 303 (2005), no. 1,242–257. doi: 10.1016/j.jmaa.2004.08.039
    [28] Y. Wang, A sufficient condition for finite time blow up of the nonlinear Klein-Gordon equations with arbitrarily positive initial energy, Proc. Amer. Math. Soc., 136 (2008), no. 10, 3477–3482. doi: 10.1090/S0002-9939-08-09514-2
    [29] S. Wang and G. Chen, Cauchy problem of the generalized double dispersion equation, Nonlinear Anal., 64 (2006), no. 1,159–173. doi: 10.1016/j.na.2005.06.017
    [30] The Cauchy problem for the dissipative Boussinesq equation. Nonlinear Anal. Real World Appl. (2019) 45: 116-141.
    [31] Global existence and nonexistence of the initial-boundary value problem for the dissipative Boussinesq equation. Nonlinear Anal. (2016) 134: 164-188.
    [32] M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Applications, Vol. 24, Birkhäuser Boston, Inc., Boston, MA, 1996. doi: 10.1007/978-1-4612-4146-1
    [33] R. Xu, Y. Liu and T. Yu, Global existence of solution for Cauchy problem of multidimensional generalized double dispersion equations, Nonlinear Anal., 71 (2009), no. 10, 4977–4983. doi: 10.1016/j.na.2009.03.069
    [34] R. Xu and Y. Liu, Global existence and nonexistence of solution for Cauchy problem of multidimensional double dispersion equations, J. Math. Anal. Appl., 359 (2009), no. 2,739–751. doi: 10.1016/j.jmaa.2009.06.034
    [35] R. Xu, Y. Yang, B. Liu, J. Shen and S. Huang, Global existence and blowup of solutions for the multidimensional sixth-order "good" Boussinesq equation, Z. Angew. Math. Phys., 66 (2015), no. 3,955–976. doi: 10.1007/s00033-014-0459-9
    [36] R. Xu, M. Zhang, S. Chen, Y. Yang and J. Shen, The initial-boundary value problems for a class of sixth order nonlinear wave equation, Discrete Contin. Dyn. Syst., 37 (2017), no. 11, 5631–5649. doi: 10.3934/dcds.2017244
    [37] Y. Yang and R. Xu, Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up, Commun. Pure Appl. Anal., 18 (2019), no. 3, 1351–1358. doi: 10.3934/cpaa.2019065
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