American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition
  • CPAA Home
  • This Issue
  • Next Article
    A generalized complex Ginzburg-Landau equation: Global existence and stability results
May  2021, 20(5): 2039-2064. doi: 10.3934/cpaa.2021057

Global well-posedness 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 of mechanic and energetic, University of Chlef, Algeria
3. 

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

* Corresponding author

Received  October 2020 Revised  February 2021 Published  May 2021 Early access  April 2021

Fund Project: 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 global (in time) well-posedness results for 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 there appears a Fujita type exponent as a threshold. Applying modern tools from Harmonic Analysis we prove several results by taking into consideration different regularity properties of the data.

Keywords: Damped wave equation, effective damping, nonlinear memory, Cauchy problem, energy solutions.
Mathematics Subject Classification: Primary: 35G25, 35B40; Secondary: 35B33.
Citation: Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, 2021, 20 (5) : 2039-2064. doi: 10.3934/cpaa.2021057
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]

S. Cui, Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal., 43 (2001), 293-323.  doi: 10.1016/S0362-546X(99)00195-9.  Google Scholar

[3]

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

[4]

M. D'Abbicco, G. Girardi and M. Reissig, A scale of critical exponents for semilinear waves with time-dependent damping and mass terms, Nonlinear Anal., 179 (2019), 15-40.  doi: 10.1016/j.na.2018.08.006.  Google Scholar

[5]

M. D'Abbicco and S. Lucente, The beam equation with nonlinear memory, Z. Angew. Math. Phys., 67 (2016), 18 pp. doi: 10.1007/s00033-016-0655-x.  Google Scholar

[6]

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

[7]

A. Djaouti and M. Reissig, Weakly 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

[8]

M. R. Ebert and M. Reissig, Methods for Partial Differential Equations. Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, Birkhäuser, Cham, 2018. doi: 10.1007/978-3-319-66456-9.  Google Scholar

[9]

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

[10]

T. Hadj Kaddour and M. Reissig, Blow-up results for effectively damped wave models with nonlinear memory, 21 pp., accepted for publication in CPAA. Google Scholar

[11]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter series in nonlinear analysis and applications, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411.  Google Scholar

[12]

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]

S. Cui, Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal., 43 (2001), 293-323.  doi: 10.1016/S0362-546X(99)00195-9.  Google Scholar

[3]

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

[4]

M. D'Abbicco, G. Girardi and M. Reissig, A scale of critical exponents for semilinear waves with time-dependent damping and mass terms, Nonlinear Anal., 179 (2019), 15-40.  doi: 10.1016/j.na.2018.08.006.  Google Scholar

[5]

M. D'Abbicco and S. Lucente, The beam equation with nonlinear memory, Z. Angew. Math. Phys., 67 (2016), 18 pp. doi: 10.1007/s00033-016-0655-x.  Google Scholar

[6]

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

[7]

A. Djaouti and M. Reissig, Weakly 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

[8]

M. R. Ebert and M. Reissig, Methods for Partial Differential Equations. Qualitative Properties of Solutions, Phase Space Analysis, Semilinear Models, Birkhäuser, Cham, 2018. doi: 10.1007/978-3-319-66456-9.  Google Scholar

[9]

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

[10]

T. Hadj Kaddour and M. Reissig, Blow-up results for effectively damped wave models with nonlinear memory, 21 pp., accepted for publication in CPAA. Google Scholar

[11]

T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations, De Gruyter series in nonlinear analysis and applications, Walter de Gruyter & Co., Berlin, 1996. doi: 10.1515/9783110812411.  Google Scholar

[12]

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]

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
[2]

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
[3]

Mohammad A. Rammaha, Daniel Toundykov, Zahava Wilstein. Global existence and decay of energy for a nonlinear wave equation with $p$-Laplacian damping. Discrete & Continuous Dynamical Systems, 2012, 32 (12) : 4361-4390. doi: 10.3934/dcds.2012.32.4361
[4]

Moez Daoulatli. Energy decay rates for solutions of the wave equation with linear damping in exterior domain. Evolution Equations & Control Theory, 2016, 5 (1) : 37-59. doi: 10.3934/eect.2016.5.37
[5]

Marcello D'Abbicco. Small data solutions for semilinear wave equations with effective damping. Conference Publications, 2013, 2013 (special) : 183-191. doi: 10.3934/proc.2013.2013.183
[6]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations & Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017
[7]

Kim Dang Phung. Decay of solutions of the wave equation with localized nonlinear damping and trapped rays. Mathematical Control & Related Fields, 2011, 1 (2) : 251-265. doi: 10.3934/mcrf.2011.1.251
[8]

Hiroshi Takeda. Large time behavior of solutions for a nonlinear damped wave equation. Communications on Pure & Applied Analysis, 2016, 15 (1) : 41-55. doi: 10.3934/cpaa.2016.15.41
[9]

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
[10]

Tayeb Hadj Kaddour, Michael Reissig. Blow-up results for effectively damped wave models with nonlinear memory. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2687-2707. doi: 10.3934/cpaa.2020239
[11]

Fabrizio Colombo, Davide Guidetti. Identification of the memory kernel in the strongly damped wave equation by a flux condition. Communications on Pure & Applied Analysis, 2009, 8 (2) : 601-620. doi: 10.3934/cpaa.2009.8.601
[12]

Dalibor Pražák. On the dimension of the attractor for the wave equation with nonlinear damping. Communications on Pure & Applied Analysis, 2005, 4 (1) : 165-174. doi: 10.3934/cpaa.2005.4.165
[13]

Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021093
[14]

Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090
[15]

Stéphane Gerbi, Belkacem Said-Houari. Exponential decay for solutions to semilinear damped wave equation. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 559-566. doi: 10.3934/dcdss.2012.5.559
[16]

Linglong Du, Caixuan Ren. Pointwise wave behavior of the initial-boundary value problem for the nonlinear damped wave equation in $\mathbb{R}_{+}^{n} $. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3265-3280. doi: 10.3934/dcdsb.2018319
[17]

Goro Akagi. Energy solutions of the Cauchy-Neumann problem for porous medium equations. Conference Publications, 2009, 2009 (Special) : 1-10. doi: 10.3934/proc.2009.2009.1
[18]

M. Nakamura, Tohru Ozawa. The Cauchy problem for nonlinear wave equations in the Sobolev space of critical order. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 215-231. doi: 10.3934/dcds.1999.5.215
[19]

Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024
[20]

Vanessa Barros, Carlos Nonato, Carlos Raposo. Global existence and energy decay of solutions for a wave equation with non-constant delay and nonlinear weights. Electronic Research Archive, 2020, 28 (1) : 205-220. doi: 10.3934/era.2020014

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (139)
  • HTML views (124)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy