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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 |
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.
References:
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An equation whose Fujita critical exponent is not given by scaling, Nonlinear Anal., 68 (2008), 862-874.
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M. D'Abbicco,
The influence of a nonlinear memory on the damped wave equation, Nonlinear Anal., 95 (2014), 130-145.
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M. D'Abbicco, S. Lucente and M. Reissig,
Semilinear wave equations with effective damping, Chin. Ann. Math., Serie B, 34 (2013), 345-380.
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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.
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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. |
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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. |
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S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. |
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J. Wirth,
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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] |
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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
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