American Institute of Mathematical Sciences

Advanced
doi: 10.3934/cpaa.2020239

Blow-up results for effectively damped wave models with nonlinear memory

Tayeb Hadj Kaddour 1,2, and  Michael Reissig 3,,

1. 

Department of Mathematics, Faculty of exact sciences and informatics, University of Chlef, P. O. Box 50, 02000, Ouled-Fares, Chlef, Algeria
2. 

Laboratory ACEDP, Djillali Liabes University, 22000 Sidi Bel Abbes, Algeria
3. 

Faculty for Mathematics and Computer Science, TU Bergakademie Freiberg, Prüferstr. 9, 09596, Freiberg, Germany

* Corresponding author

Received  April 2020 Revised  May 2020 Published  September 2020

Fund Project: In honor of Prof. Chen Shuxing on ocassion of his 80th birthday. The research of this paper is supported by DAAD, Erasmus+ Project between the Hassiba Benbouali University of Chlef and TU Bergakademie Freiberg, 2015-1-DE01-KA107-002026, during the stay of the first author at Technical University Bergakademie Freiberg within the periods April 2016 to June 2016, and a stay of one month April 2017 supported by Hassiba Benbouali University

In this paper, we study the Cauchy problem for a special family of effectively damped wave models with nonlinear memory on the right-hand side. Our goal is to prove blow-up results for local (in time) Sobolev solutions. Due to the effective dissipation the model is parabolic like from the point of view of energy decay estimates of the corresponding linear Cauchy problem with vanishing right-hand side. For this reason we apply the test function method for proving our results.

Keywords: Damped wave equation, effective damping, nonlinear memory, Cauchy problem, energy solutions, blow-up, test function method.
Mathematics Subject Classification: Primary: 35G25, 35B40; Secondary: 35B33, 35C20.
Citation: Tayeb Hadj Kaddour, Michael Reissig. Blow-up results for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, doi: 10.3934/cpaa.2020239
References:
[1]

T. Cazanave, F. Dickstein and F. D. Weissler, An equation whose Fujita critical exponent is not given by scaling, Nonlinear Anal., 68 (2008), 862-874.  doi: 10.1016/j.na.2006.11.042.  Google Scholar

[2]

M. D'Abbicco, The influence of a nonlinear memory on the damped wave equation, Nonlinear Anal., 95 (2014), 130-145.  doi: 10.1016/j.na.2013.09.006.  Google Scholar

[3]

M. D'Abbicco, S. Lucente and M. Reissig, Semilinear wave equations with effective damping, Chin. Ann. Math., Serie B, 34 (2013), 345-380.  doi: 10.1007/s11401-013-0773-0.  Google Scholar

[4]

I. Dannawi, M. Kirane and A. Fino, Finite time blow-up for damped wave equations with space-time dependent potential and nonlinear memory, Nonlinear Differ. Equ. Appl., 25 (2018), 19 pp. doi: 10.1007/s00030-018-0533-7.  Google Scholar

[5]

A. Djaouti and M. Reissig, Coupled systems of semilinear effectively damped waves with time-dependent coefficient, different power nonlinearities and different regularity of the data, Nonlinear Anal., 175 (2018), 28-55.  doi: 10.1016/j.na.2018.05.006.  Google Scholar

[6]

A. Fino, Critical exponent for damped wave equations with nonlinear memory, Nonlinear Anal., 74 (2011), 5495-5505.  doi: 10.1016/j.na.2011.01.039.  Google Scholar

[7]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[8]

J. Wirth, Wave equations with time-dependent dissipation Ⅱ. Effective dissipation, J. Differ. Equ., 232 (2007), 74-103.  doi: 10.1016/j.jde.2006.06.004.  Google Scholar

show all references

References:
[1]

T. Cazanave, F. Dickstein and F. D. Weissler, An equation whose Fujita critical exponent is not given by scaling, Nonlinear Anal., 68 (2008), 862-874.  doi: 10.1016/j.na.2006.11.042.  Google Scholar

[2]

M. D'Abbicco, The influence of a nonlinear memory on the damped wave equation, Nonlinear Anal., 95 (2014), 130-145.  doi: 10.1016/j.na.2013.09.006.  Google Scholar

[3]

M. D'Abbicco, S. Lucente and M. Reissig, Semilinear wave equations with effective damping, Chin. Ann. Math., Serie B, 34 (2013), 345-380.  doi: 10.1007/s11401-013-0773-0.  Google Scholar

[4]

I. Dannawi, M. Kirane and A. Fino, Finite time blow-up for damped wave equations with space-time dependent potential and nonlinear memory, Nonlinear Differ. Equ. Appl., 25 (2018), 19 pp. doi: 10.1007/s00030-018-0533-7.  Google Scholar

[5]

A. Djaouti and M. Reissig, Coupled systems of semilinear effectively damped waves with time-dependent coefficient, different power nonlinearities and different regularity of the data, Nonlinear Anal., 175 (2018), 28-55.  doi: 10.1016/j.na.2018.05.006.  Google Scholar

[6]

A. Fino, Critical exponent for damped wave equations with nonlinear memory, Nonlinear Anal., 74 (2011), 5495-5505.  doi: 10.1016/j.na.2011.01.039.  Google Scholar

[7]

S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993.  Google Scholar

[8]

J. Wirth, Wave equations with time-dependent dissipation Ⅱ. Effective dissipation, J. Differ. Equ., 232 (2007), 74-103.  doi: 10.1016/j.jde.2006.06.004.  Google Scholar

[1]

Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831
[2]

Yanbing Yang, Runzhang Xu. Nonlinear wave equation with both strongly and weakly damped terms: Supercritical initial energy finite time blow up. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1351-1358. doi: 10.3934/cpaa.2019065
[3]

Jorge A. Esquivel-Avila. Blow-up in damped abstract nonlinear equations. Electronic Research Archive, 2020, 28 (1) : 347-367. doi: 10.3934/era.2020020
[4]

Gongwei Liu. The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electronic Research Archive, 2020, 28 (1) : 263-289. doi: 10.3934/era.2020016
[5]

Jianbo Cui, Jialin Hong, Liying Sun. On global existence and blow-up for damped stochastic nonlinear Schrödinger equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6837-6854. doi: 10.3934/dcdsb.2019169
[6]

Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006
[7]

Victor A. Galaktionov, Juan-Luis Vázquez. The problem Of blow-up in nonlinear parabolic equations. Discrete & Continuous Dynamical Systems - A, 2002, 8 (2) : 399-433. doi: 10.3934/dcds.2002.8.399
[8]

Van Duong Dinh. Blow-up criteria for linearly damped nonlinear Schrödinger equations. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020082
[9]

Pierre Garnier. Damping to prevent the blow-up of the korteweg-de vries equation. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1455-1470. doi: 10.3934/cpaa.2017069
[10]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147
[11]

Claudianor O. Alves, M. M. Cavalcanti, Valeria N. Domingos Cavalcanti, Mohammad A. Rammaha, Daniel Toundykov. On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 583-608. doi: 10.3934/dcdss.2009.2.583
[12]

Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280
[13]

Satyanad Kichenassamy. Control of blow-up singularities for nonlinear wave equations. Evolution Equations & Control Theory, 2013, 2 (4) : 669-677. doi: 10.3934/eect.2013.2.669
[14]

Lorena Bociu, Petronela Radu. Existence of weak solutions to the Cauchy problem of a semilinear wave equation with supercritical interior source and damping. Conference Publications, 2009, 2009 (Special) : 60-71. doi: 10.3934/proc.2009.2009.60
[15]

Alexander Gladkov. Blow-up problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2053-2068. doi: 10.3934/cpaa.2017101
[16]

Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020388
[17]

Van Duong Dinh. On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 689-708. doi: 10.3934/cpaa.2019034
[18]

Binhua Feng. On the blow-up solutions for the fractional nonlinear Schrödinger equation with combined power-type nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1785-1804. doi: 10.3934/cpaa.2018085
[19]

Takashi Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Conference Publications, 2009, 2009 (Special) : 592-601. doi: 10.3934/proc.2009.2009.592
[20]

Mohamed Jleli, Bessem Samet. Blow-up for semilinear wave equations with time-dependent damping in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3885-3900. doi: 10.3934/cpaa.2020143

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (66)
  • HTML views (64)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences