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March  2020, 28(1): 347-367. doi: 10.3934/era.2020020

Blow-up in damped abstract nonlinear equations

Universidad Autónoma Metropolitana, Unidad Azcapotzalco, Av. San Pablo 180, Col. Reynosa Tamaulipas, 02200 Azcapotzalco, CDMX, México

* Corresponding author: Jorge A. Esquivel-Avila, jaea@azc.uam.mx

Received  December 2019 Revised  February 2020 Published  March 2020

Fund Project: This author was supported by the Universidad Autónoma Metropolitana, Unidad Azcapotzalco

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. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020
References:
[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.  Google Scholar

[2]

J. V. Boussinesq, 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., 17 (1872), 55-108.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[23] G. A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford University Press, Oxford, UK, 1999.   Google Scholar
[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.  Google Scholar

[25]

N. Polat and A. Ertaș, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl., 349 (2009), 10-20.  doi: 10.1016/j.jmaa.2008.08.025.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

S. Wang and X. Su, The Cauchy problem for the dissipative Boussinesq equation, Nonlinear Anal. Real World Appl., 45 (2019), 116-141.  doi: 10.1016/j.nonrwa.2018.06.012.  Google Scholar

[31]

S. Wang and X. Su, Global existence and nonexistence of the initial-boundary value problem for the dissipative Boussinesq equation, Nonlinear Anal., 134 (2016), 164-188.  doi: 10.1016/j.na.2016.01.004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[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.  Google Scholar

[2]

J. V. Boussinesq, 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., 17 (1872), 55-108.   Google Scholar

[3]

I. C. ChristoG. A. Maugin and A. V. Porubov, On Boussinesq's paradigm in nonlinear wave propagation, Comptes Rendus Mécanique, 335 (2007), 521-535.  doi: 10.1016/j.crme.2007.08.006.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.  doi: 10.1016/j.anihpc.2005.02.007.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23] G. A. Maugin, Nonlinear Waves in Elastic Crystals, Oxford University Press, Oxford, UK, 1999.   Google Scholar
[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.  Google Scholar

[25]

N. Polat and A. Ertaș, Existence and blow-up of solution of Cauchy problem for the generalized damped multidimensional Boussinesq equation, J. Math. Anal. Appl., 349 (2009), 10-20.  doi: 10.1016/j.jmaa.2008.08.025.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

S. Wang and X. Su, The Cauchy problem for the dissipative Boussinesq equation, Nonlinear Anal. Real World Appl., 45 (2019), 116-141.  doi: 10.1016/j.nonrwa.2018.06.012.  Google Scholar

[31]

S. Wang and X. Su, Global existence and nonexistence of the initial-boundary value problem for the dissipative Boussinesq equation, Nonlinear Anal., 134 (2016), 164-188.  doi: 10.1016/j.na.2016.01.004.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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