Applications of Mathematics, Vol. 68, No. 2, pp. 209-254, 2023
Continuous dependence and general decay of solutions for a wave equation
with a nonlinear memory term
Doan Thi Nhu Quynh, Nguyen Huu Nhan, Le Thi Phuong Ngoc, Nguyen Thanh Long
Received September 12, 2021. Published online February 8, 2022.
Abstract: We study existence, uniqueness, continuous dependence, general decay of solutions of an initial boundary value problem for a viscoelastic wave equation with strong damping and nonlinear memory term. At first, we state and prove a theorem involving local existence and uniqueness of a weak solution. Next, we establish a sufficient condition to get an estimate of the continuous dependence of the solution with respect to the kernel function and the nonlinear terms. Finally, under suitable conditions to obtain the global solution, we prove the general decay property with positive initial energy for this global solution.
Keywords: viscoelastic equations; strong damping; nonlinear memory; general decay
References: [1] S. Bayraktar, Ş. Gür: Continuous dependence of solutions for damped improved Boussinesq equation. Turk. J. Math. 44 (2020), 334-341. DOI 10.3906/mat-1912-20 | MR 4059543 | Zbl 1450.35042
[2] P. Benilan, M. G. Crandall: The continuous dependence on $\phi$ of solutions of $u_t-\Delta_{\phi}(u)=0$. Indiana Univ. Math. J. 30 (1981), 161-177. DOI 10.1512/iumj.1981.30.30014 | MR 0604277 | Zbl 0482.35012
[3] N. Boumaza, S. Boulaaras: General decay for Kirchhoff type in viscoelasticity with not necessarily decreasing kernel. Math. Methods Appl. Sci. 41 (2018), 6050-6069. DOI 10.1002/mma.5117 | MR 3879228 | Zbl 1415.35038
[4] M. M. Cavalcanti, V. N. Domingos Cavalcanti, P. Martinez: General decay rate estimates for viscoelastic dissipative systems. Nonlinear Anal., Theory Methods Appl., Ser. A 68 (2008), 177-193. DOI 10.1016/j.na.2006.10.040 | MR 2361147 | Zbl 1124.74009
[5] B. Cockburn, G. Gripenber: Continuous dependence on the nonlinearities of solutions of degenerate parabolic equations. J. Differ. Equations 151 (1999), 231-251. DOI 10.1006/jdeq.1998.3499 | MR 1669570 | Zbl 0921.35017
[6] M. Conti, V. Pata: General decay properties of abstract linear viscoelasticity. Z. Angew. Math. Phys. 71 (2020), Article ID 6, 21 pages. DOI 10.1007/s00033-019-1229-5 | MR 4041122 | Zbl 1430.35030
[7] M. D'Abbicco: The influence of a nonlinear memory on the damped wave equation. Nonlinear Anal., Theory Methods Appl., Ser. A 95 (2014), 130-145. DOI 10.1016/j.na.2013.09.006 | MR 3130512 | Zbl 1284.35286
[8] M. D'Abbicco, S. Lucente: The beam equation with nonlinear memory. Z. Angew. Math. Phys. 67 (2016), Article ID 60, 18 pages. DOI 10.1007/s00033-016-0655-x | MR 3493963 | Zbl 1361.35116
[9] A. Douglis: The continuous dependence of generalized solutions of non-linear partial differential equations upon initial data. Commun. Pure Appl. Math. 14 (1961), 267-284. DOI 10.1002/cpa.3160140307 | MR 0139848 | Zbl 0117.31102
[10] G. Duvaut, J. L. Lions: Inequalities in Mechanics and Physics. Grundlehren der mathematischen Wissenschaften 219. Springer, Berlin (1976). DOI 10.1007/978-3-642-66165-5 | MR 0521262 | Zbl 0331.35002
[11] F. Ekinci, E. Pişkin, S. M. Boulaaras, I. Mekawy: Global existence and general decay of solutions for a quasilinear system with degenerate damping terms. J. Funct. Spaces 2021 (2021), Article ID 4316238, 10 pages. DOI 10.1155/2021/4316238 | MR 4283631 | Zbl 1472.35239
[12] A. Z. Fino: Critical exponent for damped wave equations with nonlinear memory. Nonlinear Anal., Theory Methods Appl., Ser. A 74 (2011), 5495-5505. DOI 10.1016/j.na.2011.01.039 | MR 2819292 | Zbl 1222.35025
[13] G. Gripenberg: Global existence of solutions of Volterra integrodifferential equations of parabolic type. J. Differ. Equations 102 (1993), 382-390. DOI 10.1006/jdeq.1993.1035 | MR 1216735 | Zbl 0780.45012
[14] Ş. Gür, M. E. Uysal: Continuous dependence of solutions to the strongly damped nonlinear Klein-Gordon equation. Turk. J. Math. 42 (2018), 904-910. DOI 10.3906/mat-1706-30 | MR 3804959 | Zbl 1424.35261
[15] X. Han, M. Wang: General decay of energy for a viscoelastic equation with nonlinear damping. J. Franklin Inst. 347 (2010), 806-817. DOI 10.1016/j.jfranklin.2010.02.010 | MR 2645392 | Zbl 1286.35148
[16] J. Hao, H. Wei: Blow-up and global existence for solution of quasilinear viscoelastic wave equation with strong damping and source term. Bound. Value Probl. 2017 (2017), Article ID 65, 12 pages. DOI 10.1186/s13661-017-0796-7 | MR 3647200 | Zbl 1379.35192
[17] J. H. Hassan, S. A. Messaoudi: General decay results for a viscoelastic wave equation with a variable exponent nonlinearity. Asymptotic Anal. 125 (2021), 365-388. DOI 10.3233/ASY-201661
[18] W. J. Hrusa: Global existence and asymptotic stability for a semilinear hyperbolic Volterra equation with large initial data. SIAM J. Math. Anal. 16 (1985), 110-134. DOI 10.1137/0516007 | MR 0772871 | Zbl 0571.45007
[19] M. Jleli, B. Samet, C. Vetro: Large time behavior for inhomogeneous damped wave equations with nonlinear memory. Symmetry 12 (2020), Article ID 1609, 12 pages. DOI 10.3390/sym12101609
[20] F. John: Continuous dependence on data for solutions of partial differential equations with a prescribed bound. Commun. Pure Appl. Math. 13 (1960), 551-586. DOI 10.1002/cpa.3160130402 | MR 130456 | Zbl 0097.08101
[21] T. H. Kaddour, M. Reissig: Global well-posedness for effectively damped wave models with nonlinear memory. Commun. Pure Appl. Anal. 20 (2021), 2039-2064. DOI 10.3934/cpaa.2021057 | MR 4259639 | Zbl 1466.35264
[22] M. Kafini, S. A. Messaoudi: A blow-up result in a Cauchy viscoelastic problem. Appl. Math. Lett. 21 (2008), 549-553. DOI 10.1016/j.aml.2007.07.004 | MR 2412376 | Zbl 1149.35076
[23] M. Kafini, M. I. Mustafa: Blow-up result in a Cauchy viscoelastic problem with strong damping and dispersive. Nonlinear Anal., Real World Appl. 20 (2014), 14-20. DOI 10.1016/j.nonrwa.2014.04.005 | MR 3233895 | Zbl 1295.35129
[24] Q. Li, L. He: General decay and blow-up of solutions for a nonlinear viscoelastic wave equation with strong damping. Bound. Value Probl. 2018 (2018), Article ID 153, 22 pages. DOI 10.1186/s13661-018-1072-1 | MR 3859565
[25] J. L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Etudes mathematiques. Dunod, Paris (1969). (In French.) MR 0259693 | Zbl 0189.40603
[26] N. T. Long, A. P. N. Dinh, L. X. Truong: Existence and decay of solutions of a nonlinear viscoelastic problem with a mixed nonhomogeneous condition. Numer. Funct. Anal. Optim. 29 (2008), 1363-1393. DOI 10.1080/01630560802605955 | MR 2479113 | Zbl 1162.35053
[27] F. Mesloub, S. Boulaaras: General decay for a viscoelastic problem with not necessarily decreasing kernel. J. Appl. Math. Comput. 58 (2018), 647-665. DOI 10.1007/s12190-017-1161-9 | MR 3847059 | Zbl 1403.35050
[28] S. A. Messaoudi: Blow up and global existence in a nonlinear viscoelastic wave equation. Math. Nachr. 260 (2003), 58-66. DOI 10.1002/mana.200310104 | MR 2017703 | Zbl 1035.35082
[29] S. A. Messaoudi: General decay of the solution energy in a viscoelastic equation with a nonlinear source. Nonlinear Anal., Theory Methods Appl., Ser. A 69 (2008), 2589-2598. DOI 10.1016/j.na.2007.08.035 | MR 2446355 | Zbl 1154.35066
[30] M. I. Mustafa: General decay result for nonlinear viscoelastic equations. J. Math. Anal. Appl. 457 (2018), 134-152. DOI 10.1016/j.jmaa.2017.08.019 | MR 3702699 | Zbl 1379.35028
[31] M. I. Mustafa: Optimal decay rates for the viscoelastic wave equation. Math. Methods Appl. Sci. 41 (2018), 192-204. DOI 10.1002/mma.4604 | MR 3745365 | Zbl 1391.35058
[32] L. T. P. Ngoc, D. T. N. Quynh, N. A. Triet, N. T. Long: Linear approximation and asymptotic expansion associated to the Robin-Dirichlet problem for a Kirchhoff-Carrier equation with a viscoelastic term. Kyungpook Math. J. 59 (2019), 735-769. MR 4057771
[33] J. Q. Pan: The continuous dependence on nonlinearities of solutions of the Neumann problem of a singular parabolic equation. Nonlinear Anal., Theory Methods Appl., Ser. A 67 (2007), 2081-2090. DOI 10.1016/j.na.2006.09.017 | MR 2331859 | Zbl 1123.35026
[34] D. T. N. Quynh, B. D. Nam, L. T. M. Thanh, T. T. M. Dung, N. H. Nhan: High-order iterative scheme for a viscoelastic wave equation and numerical results. Math. Probl. Eng. 2021 (2021), Article ID 9917271, 27 pages. DOI 10.1155/2021/9917271 | MR 4274176
[35] Y. Shang, B. Guo: On the problem of the existence of global solutions for a class of nonlinear convolutional intergro-differential equations of pseudoparabolic type. Acta Math. Appl. Sin. 26 (2003), 511-524. (In Chinese.) MR 2022221 | Zbl 1057.45004
Affiliations: Doan Thi Nhu Quynh, Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam; Vietnam National University, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam; Faculty of Applied Sciences, Ho Chi Minh City University of Food Industry, 140 Le Trong Tan Str., Tay Thanh Ward, Tan Phu Dist., Ho Chi Minh City, Vietnam, e-mail: doanthinhuquynh02@gmail.com; Nguyen Huu Nhan, Nguyen Tat Thanh University, 300A Nguyen Tat Thanh Str., Dist. 4, Ho Chi Minh City, Vietnam, e-mail: huunhandn@gmail.com; Le Thi Phuong Ngoc, University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang City, Vietnam, e-mail: ngoc1966@gmail.com; Nguyen Thanh Long (corresponding author), Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, Vietnam; Vietnam National University, Linh Trung Ward, Thu Duc City, Ho Chi Minh City, Vietnam, e-mail: longnt2@gmail.com
