# American Institute of Mathematical Sciences

doi: 10.3934/cpaa.2020239

## Blow-up results 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 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.

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
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##### 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
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