| Title:
|
On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values (English) |
| Author:
|
Dzung, Nguyen Vu |
| Author:
|
Ngoc, Le Thi Phuong |
| Author:
|
Nhan, Nguyen Huu |
| Author:
|
Long, Nguyen Thanh |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
149 |
| Issue:
|
2 |
| Year:
|
2024 |
| Pages:
|
261-285 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values $u(\eta _{1},t),\cdots ,u(\eta _{q},t)$ with $0\leq \eta _{1}<\eta _{2}<\cdots <\eta _{q}<1.$ By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case $({\rm P}_{q})$ of (P) in which the nonlinear term contains the sum $S_{q}[u^{2}](t)=q^{-1}\sum _{i=1}^{q}u^{2}(\frac{(i-1)}{q},t)$. Under suitable conditions, we prove that the solution of $({\rm P}_{q})$ converges to the solution of the corresponding problem $({\rm P}_{\infty })$ as $q\rightarrow \infty $ (in a certain sense), here $({\rm P}_{\infty })$ is defined by $({\rm P}_{q})$ in which $S_{q}[u^{2}](t)$ is replaced by $ \int _{0}^{1}u^{2}( y,t) {\rm d}y.$ The proof is done by using the compactness lemma of Aubin-Lions and the method of continuity with a priori estimates. We end the paper with remarks related to similar problems. (English) |
| Keyword:
|
Kirchhoff-Carrier equation |
| Keyword:
|
Robin-Dirichlet problem |
| Keyword:
|
nonlocal term |
| Keyword:
|
Faedo-Galerkin method |
| Keyword:
|
linearization method |
| MSC:
|
35A01 |
| MSC:
|
35A02 |
| MSC:
|
35B45 |
| MSC:
|
35L05 |
| MSC:
|
35M11 |
| idZBL:
|
Zbl 07893423 |
| idMR:
|
MR4767012 |
| DOI:
|
10.21136/MB.2023.0153-21 |
| . |
| Date available:
|
2024-07-10T15:06:35Z |
| Last updated:
|
2024-12-13 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/152472 |
| . |
| Reference:
|
[1] Agarwal, R. P.: Boundary Value Problems for Higher Order Differential Equations.World Scientific, Singapore (1986). Zbl 0619.34019, MR 1021979, 10.1142/0266 |
| Reference:
|
[2] Andreu-Vaillo, F., Mazón, J. M., Rossi, J. D., Toledo-Melero, J. J.: Nonlocal Diffusion Problems.Mathematical Surveys and Monographs 165. AMS, Providence (2010). Zbl 1214.45002, MR 2722295, 10.1090/surv/165 |
| Reference:
|
[3] Carrier, G. F.: On the non-linear vibration problem of the elastic string.Q. Appl. Math. 3 (1945), 157-165. Zbl 0063.00715, MR 0012351, 10.1090/qam/12351 |
| Reference:
|
[4] Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Filho, J. S. Prates, Soriano, J. A.: Existence and exponential decay for a Kirchhoff-Carrier model with viscosity.J. Math. Anal. Appl. 226 (1998), 40-60. Zbl 0914.35081, MR 1646453, 10.1006/jmaa.1998.6057 |
| Reference:
|
[5] Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Soriano, J. A.: Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation.Adv. Differ. Equ. 6 (2001), 701-730. Zbl 1007.35049, MR 1829093, 10.57262/ade/1357140586 |
| Reference:
|
[6] Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Soriano, J. A., Filho, J. S. Prates: Existence and asymptotic behaviour for a degenerate Kirchhoff-Carrier model with viscosity and nonlinear boundary conditions.Rev. Mat. Complut. 14 (2001), 177-203. Zbl 0983.35025, MR 1851728, 10.5209/rev_REMA.2001.v14.n1.17054 |
| Reference:
|
[7] Kafini, M., Messaoudi, S. A.: A blow-up result in a Cauchy viscoelastic problem.Appl. Math. Lett. 21 (2008), 549-553. Zbl 1149.35076, MR 2412376, 10.1016/j.aml.2007.07.004 |
| Reference:
|
[8] Kafini, M., Mustafa, M. I.: Blow-up result in a Cauchy viscoelastic problem with strong damping and dispersive.Nonlinear Anal., Real World Appl. 20 (2014), 14-20. Zbl 1295.35129, MR 3233895, 10.1016/j.nonrwa.2014.04.005 |
| Reference:
|
[9] Kirchhoff, G.: Vorlesungen über mathematische Physik. Erster Band. Mechanik.B. G. Teubner, Leipzig (1897), German \99999JFM99999 28.0603.01. MR 1546520 |
| Reference:
|
[10] Larkin, N. A.: Global regular solutions for the nonhomogeneous Carrier equation.Math. Probl. Eng. 8 (2002), 15-31. Zbl 1051.35042, MR 1918087, 10.1080/10241230211382 |
| Reference:
|
[11] Li, Q., He, L.: 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. Zbl 1499.35099, MR 3859565, 10.1186/s13661-018-1072-1 |
| Reference:
|
[12] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires.Dunod, Gauthier-Villars, Paris (1969), French. Zbl 0189.40603, MR 0259693 |
| Reference:
|
[13] Long, N. T., Dinh, A. P. N., Diem, T. N.: Linear recursive schemes and asymptotic expansion associated with the Kirchhoff-Carrier operator.J. Math. Anal. Appl. 267 (2002), 116-134. Zbl 1004.35095, MR 1886820, 10.1006/jmaa.2001.7755 |
| Reference:
|
[14] Medeiros, L. A.: On some nonlinear perturbation of Kirchhoff-Carrier operator.Comput. Appl. Math. 13 (1994), 225-233. Zbl 0821.35100, MR 1326759 |
| Reference:
|
[15] Medeiros, L. A., Limaco, J., Menezes, S. B.: Vibrations of elastic strings: Mathematical aspects. I.J. Comput. Anal. Appl. 4 (2002), 91-127. Zbl 1118.35335, MR 1875347, 10.1023/A:1012934900316 |
| Reference:
|
[16] Medeiros, L. A., Limaco, J., Menezes, S. B.: Vibrations of elastic strings: Mathematical aspects. II.J. Comput. Anal. Appl. 4 (2002), 211-263. Zbl 1118.35336, MR 1878996, 10.1023/A:1013151525487 |
| Reference:
|
[17] Messaoudi, S. A.: Blow up and global existence in a nonlinear viscoelastic wave equation.Math. Nachr. 260 (2003), 58-66. Zbl 1035.35082, MR 2017703, 10.1002/mana.200310104 |
| Reference:
|
[18] Messaoudi, S. A.: 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. Zbl 1154.35066, MR 2446355, 10.1016/j.na.2007.08.035 |
| Reference:
|
[19] Mustafa, M. I.: General decay result for nonlinear viscoelastic equations.J. Math. Anal. Appl. 457 (2018), 134-152. Zbl 1379.35028, MR 3702699, 10.1016/j.jmaa.2017.08.019 |
| Reference:
|
[20] Nhan, N. H., Ngoc, L. T. P., Long, N. T.: Existence and asymptotic expansion of the weak solution for a wave equation with nonlinear source containing nonlocal term.Bound. Value Probl. 2017 (2017), Article ID 87, 20 pages. Zbl 1370.35212, MR 3660353, 10.1186/s13661-017-0818-5 |
| Reference:
|
[21] Park, J. Y., Bae, J. J.: On coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term.Appl. Math. Comput. 129 (2002), 87-105. Zbl 1032.35139, MR 1897321, 10.1016/S0096-3003(01)00031-5 |
| Reference:
|
[22] Park, J. Y., Park, S. H.: General decay for quasilinear viscoelastic equations with nonlinear weak damping.J. Math. Phys. 50 (2009), Article ID 083505, 10 pages. Zbl 1298.35221, MR 2554433, 10.1063/1.3187780 |
| Reference:
|
[23] Santos, M. L., Ferreira, J., Pereira, D. C., Raposo, C. A.: Global existence and stability for wave equation of Kirchhoff type with memory condition at the boundary.Nonlinear Anal., Theory Methods Appl., Ser. A 54 (2003), 959-976. Zbl 1032.35140, MR 1992515, 10.1016/S0362-546X(03)00121-4 |
| Reference:
|
[24] Showalter, R. E.: Hilbert Space Methods for Partial Differential Equations.Electronic Journal of Differential Equations. Monograph 1. Southwest Texas State University, San Marcos (1994). Zbl 0991.35001, MR 1302484 |
| Reference:
|
[25] Tatar, N., Zaraï, A.: Exponential stability and blow up for a problem with Balakrishnan-Taylor damping.Demonstr. Math. 44 (2011), 67-90. Zbl 1227.35074, MR 2796763, 10.1515/dema-2013-0297 |
| Reference:
|
[26] Wang, Y., Wang, Y.: Exponential energy decay of solutions of viscoelastic wave equations.J. Math. Anal. Appl. 347 (2008), 18-25. Zbl 1149.35323, MR 2433821, 10.1016/j.jmaa.2008.05.098 |
| Reference:
|
[27] Wu, S.-T.: Exponential energy decay of solutions for an integro-differential equation with strong damping.J. Math. Anal. Appl. 364 (2010), 609-617. Zbl 1205.45012, MR 2576211, 10.1016/j.jmaa.2009.11.046 |
| . |
