American Institute of Mathematical Sciences

• Previous Article
A generalized complex Ginzburg-Landau equation: Global existence and stability results
• CPAA Home
• This Issue
• Next Article
Asymptotic behaviors of solutions to a sixth-order Boussinesq equation with logarithmic nonlinearity
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  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 & Applied Analysis, 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] Hui Yang, Yuzhu Han. Initial boundary value problem for a strongly damped wave equation with a general nonlinearity. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021019 [2] Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021055 [3] Yanling Shi, Junxiang Xu. Quasi-periodic solutions for nonlinear wave equation with Liouvillean frequency. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3479-3490. doi: 10.3934/dcdsb.2020241 [4] Pengyan Ding, Zhijian Yang. Well-posedness and attractor for a strongly damped wave equation with supercritical nonlinearity on $\mathbb{R}^{N}$. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1059-1076. doi: 10.3934/cpaa.2021006 [5] Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027 [6] Jiacheng Wang, Peng-Fei Yao. On the attractor for a semilinear wave equation with variable coefficients and nonlinear boundary dissipation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021043 [7] Siqi Chen, Yong-Kui Chang, Yanyan Wei. Pseudo $S$-asymptotically Bloch type periodic solutions to a damped evolution equation. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021017 [8] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448 [9] Yuanqing Xu, Xiaoxiao Zheng, Jie Xin. New explicit and exact traveling wave solutions of (3+1)-dimensional KP equation. Mathematical Foundations of Computing, 2021  doi: 10.3934/mfc.2021006 [10] Olena Naboka. On synchronization of oscillations of two coupled Berger plates with nonlinear interior damping. Communications on Pure & Applied Analysis, 2009, 8 (6) : 1933-1956. doi: 10.3934/cpaa.2009.8.1933 [11] Hailing Xuan, Xiaoliang Cheng. Numerical analysis and simulation of an adhesive contact problem with damage and long memory. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2781-2804. doi: 10.3934/dcdsb.2020205 [12] Hailing Xuan, Xiaoliang Cheng. Numerical analysis of a thermal frictional contact problem with long memory. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021031 [13] Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3683-3708. doi: 10.3934/dcds.2021012 [14] Zhi-Min Chen, Philip A. Wilson. Stability of oscillatory gravity wave trains with energy dissipation and Benjamin-Feir instability. Discrete & Continuous Dynamical Systems - B, 2012, 17 (7) : 2329-2341. doi: 10.3934/dcdsb.2012.17.2329 [15] Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817 [16] Abderrazak Chrifi, Mostafa Abounouh, Hassan Al Moatassime. Galerkin method of weakly damped cubic nonlinear Schrödinger with Dirac impurity, and artificial boundary condition in a half-line. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021030 [17] Jianli Xiang, Guozheng Yan. The uniqueness of the inverse elastic wave scattering problem based on the mixed reciprocity relation. Inverse Problems & Imaging, 2021, 15 (3) : 539-554. doi: 10.3934/ipi.2021004 [18] Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003 [19] Guodong Wang, Bijun Zuo. Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021078 [20] Tomoyuki Tanaka, Kyouhei Wakasa. On the critical decay for the wave equation with a cubic convolution in 3D. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021048

2019 Impact Factor: 1.105

Article outline

[Back to Top]