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May  2021, 20(5): 2039-2064. doi: 10.3934/cpaa.2021057

## Global well-posedness for effectively damped wave models with nonlinear memory

 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.

Citation: Tayeb Hadj Kaddour, Michael Reissig. Global well-posedness for effectively damped wave models with nonlinear memory. Communications on Pure and 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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.

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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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.
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