On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values

Nguyen Vu Dzung; Le Thi Phuong Ngoc; Nguyen Huu Nhan; Nguyen Thanh Long

Mathematica Bohemica (2024)

  • Volume: 149, Issue: 2, page 261-285
  • ISSN: 0862-7959

Abstract

top
We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values u ( η 1 , t ) , ⋯ , u ( η q , t ) with 0 ≤ η 1 < η 2 < ⋯ < η 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 ( P q ) of (P) in which the nonlinear term contains the sum S q [ u 2 ] ( t ) = q - 1 ∑ i = 1 q u 2 ( ( i - 1 ) q , t ) . Under suitable conditions, we prove that the solution of ( P q ) converges to the solution of the corresponding problem ( P ∞ ) as q → ∞ (in a certain sense), here ( P ∞ ) is defined by ( P q ) in which S q [ u 2 ] ( t ) is replaced by ∫ 0 1 u 2 ( y , t ) 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.

How to cite

top

Dzung, Nguyen Vu, et al. "On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values." Mathematica Bohemica 149.2 (2024): 261-285. <http://eudml.org/doc/299590>.

@article{Dzung2024,
abstract = {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 $0\le \eta _\{1\}<\eta _\{2\}<\cdots <\eta _\{q\}<1.$ 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.},
author = {Dzung, Nguyen Vu, Ngoc, Le Thi Phuong, Nhan, Nguyen Huu, Long, Nguyen Thanh},
journal = {Mathematica Bohemica},
keywords = {Kirchhoff-Carrier equation; Robin-Dirichlet problem; nonlocal term; Faedo-Galerkin method; linearization method},
language = {eng},
number = {2},
pages = {261-285},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values},
url = {http://eudml.org/doc/299590},
volume = {149},
year = {2024},
}

TY - JOUR
AU - Dzung, Nguyen Vu
AU - Ngoc, Le Thi Phuong
AU - Nhan, Nguyen Huu
AU - Long, Nguyen Thanh
TI - On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values
JO - Mathematica Bohemica
PY - 2024
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 149
IS - 2
SP - 261
EP - 285
AB - 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 $0\le \eta _{1}<\eta _{2}<\cdots <\eta _{q}<1.$ 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.
LA - eng
KW - Kirchhoff-Carrier equation; Robin-Dirichlet problem; nonlocal term; Faedo-Galerkin method; linearization method
UR - http://eudml.org/doc/299590
ER -

References

top
  1. Agarwal, R. P., 10.1142/0266, World Scientific, Singapore (1986). (1986) Zbl0619.34019MR1021979DOI10.1142/0266
  2. Andreu-Vaillo, F., Mazón, J. M., Rossi, J. D., Toledo-Melero, J. J., 10.1090/surv/165, Mathematical Surveys and Monographs 165. AMS, Providence (2010). (2010) Zbl1214.45002MR2722295DOI10.1090/surv/165
  3. Carrier, G. F., 10.1090/qam/12351, Q. Appl. Math. 3 (1945), 157-165. (1945) Zbl0063.00715MR0012351DOI10.1090/qam/12351
  4. Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Filho, J. S. Prates, Soriano, J. A., 10.1006/jmaa.1998.6057, J. Math. Anal. Appl. 226 (1998), 40-60. (1998) Zbl0914.35081MR1646453DOI10.1006/jmaa.1998.6057
  5. Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Soriano, J. A., 10.57262/ade/1357140586, Adv. Differ. Equ. 6 (2001), 701-730. (2001) Zbl1007.35049MR1829093DOI10.57262/ade/1357140586
  6. Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Soriano, J. A., Filho, J. S. Prates, 10.5209/rev_REMA.2001.v14.n1.17054, Rev. Mat. Complut. 14 (2001), 177-203. (2001) Zbl0983.35025MR1851728DOI10.5209/rev_REMA.2001.v14.n1.17054
  7. Kafini, M., Messaoudi, S. A., 10.1016/j.aml.2007.07.004, Appl. Math. Lett. 21 (2008), 549-553. (2008) Zbl1149.35076MR2412376DOI10.1016/j.aml.2007.07.004
  8. Kafini, M., Mustafa, M. I., 10.1016/j.nonrwa.2014.04.005, Nonlinear Anal., Real World Appl. 20 (2014), 14-20. (2014) Zbl1295.35129MR3233895DOI10.1016/j.nonrwa.2014.04.005
  9. Kirchhoff, G., Vorlesungen über mathematische Physik. Erster Band. Mechanik, B. G. Teubner, Leipzig (1897), German 9999JFM99999 28.0603.01. (1897) MR1546520
  10. Larkin, N. A., 10.1080/10241230211382, Math. Probl. Eng. 8 (2002), 15-31. (2002) Zbl1051.35042MR1918087DOI10.1080/10241230211382
  11. Li, Q., He, L., 10.1186/s13661-018-1072-1, Bound. Value Probl. 2018 (2018), Article ID 153, 22 pages. (2018) Zbl1499.35099MR3859565DOI10.1186/s13661-018-1072-1
  12. Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Gauthier-Villars, Paris (1969), French. (1969) Zbl0189.40603MR0259693
  13. Long, N. T., Dinh, A. P. N., Diem, T. N., 10.1006/jmaa.2001.7755, J. Math. Anal. Appl. 267 (2002), 116-134. (2002) Zbl1004.35095MR1886820DOI10.1006/jmaa.2001.7755
  14. Medeiros, L. A., On some nonlinear perturbation of Kirchhoff-Carrier operator, Comput. Appl. Math. 13 (1994), 225-233. (1994) Zbl0821.35100MR1326759
  15. Medeiros, L. A., Limaco, J., Menezes, S. B., 10.1023/A:1012934900316, J. Comput. Anal. Appl. 4 (2002), 91-127. (2002) Zbl1118.35335MR1875347DOI10.1023/A:1012934900316
  16. Medeiros, L. A., Limaco, J., Menezes, S. B., 10.1023/A:1013151525487, J. Comput. Anal. Appl. 4 (2002), 211-263. (2002) Zbl1118.35336MR1878996DOI10.1023/A:1013151525487
  17. Messaoudi, S. A., 10.1002/mana.200310104, Math. Nachr. 260 (2003), 58-66. (2003) Zbl1035.35082MR2017703DOI10.1002/mana.200310104
  18. Messaoudi, S. A., 10.1016/j.na.2007.08.035, Nonlinear Anal., Theory Methods Appl., Ser. A 69 (2008), 2589-2598. (2008) Zbl1154.35066MR2446355DOI10.1016/j.na.2007.08.035
  19. Mustafa, M. I., 10.1016/j.jmaa.2017.08.019, J. Math. Anal. Appl. 457 (2018), 134-152. (2018) Zbl1379.35028MR3702699DOI10.1016/j.jmaa.2017.08.019
  20. Nhan, N. H., Ngoc, L. T. P., Long, N. T., 10.1186/s13661-017-0818-5, Bound. Value Probl. 2017 (2017), Article ID 87, 20 pages. (2017) Zbl1370.35212MR3660353DOI10.1186/s13661-017-0818-5
  21. Park, J. Y., Bae, J. J., 10.1016/S0096-3003(01)00031-5, Appl. Math. Comput. 129 (2002), 87-105. (2002) Zbl1032.35139MR1897321DOI10.1016/S0096-3003(01)00031-5
  22. Park, J. Y., Park, S. H., 10.1063/1.3187780, J. Math. Phys. 50 (2009), Article ID 083505, 10 pages. (2009) Zbl1298.35221MR2554433DOI10.1063/1.3187780
  23. Santos, M. L., Ferreira, J., Pereira, D. C., Raposo, C. A., 10.1016/S0362-546X(03)00121-4, Nonlinear Anal., Theory Methods Appl., Ser. A 54 (2003), 959-976. (2003) Zbl1032.35140MR1992515DOI10.1016/S0362-546X(03)00121-4
  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). (1994) Zbl0991.35001MR1302484
  25. Tatar, N., Zaraï, A., 10.1515/dema-2013-0297, Demonstr. Math. 44 (2011), 67-90. (2011) Zbl1227.35074MR2796763DOI10.1515/dema-2013-0297
  26. Wang, Y., Wang, Y., 10.1016/j.jmaa.2008.05.098, J. Math. Anal. Appl. 347 (2008), 18-25. (2008) Zbl1149.35323MR2433821DOI10.1016/j.jmaa.2008.05.098
  27. Wu, S.-T., 10.1016/j.jmaa.2009.11.046, J. Math. Anal. Appl. 364 (2010), 609-617. (2010) Zbl1205.45012MR2576211DOI10.1016/j.jmaa.2009.11.046

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.