| Title:
|
On a Robin-Dirichlet problem for a system of nonlinear pseudoparabolic equations with the viscoelastic term (English) |
| Author:
|
Uyen, Khong Thi Thao |
| Author:
|
Triet, Nguyen Anh |
| Author:
|
Ngoc, Le Thi Phuong |
| Author:
|
Long, Nguyen Thanh |
| Language:
|
English |
| Journal:
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Mathematica Bohemica |
| ISSN:
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0011-4642 |
| ISSN:
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0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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151 |
| Issue:
|
1 |
| Year:
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2026 |
| Pages:
|
105-152 |
| Summary lang:
|
English |
| . |
| Category:
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math |
| . |
| Summary:
|
We consider a Robin-Dirichlet problem for a system of nonlinear pseudoparabolic equations with the viscoelastic term. Based on the Faedo-Galerkin method, we first prove existence and uniqueness. Next, we give a sufficient condition for the global existence and decay of weak solutions. Finally, using concavity method, we prove blow-up results for solutions when the initial energy is nonnegative or negative. Furthermore, we establish here the lifespan for the equation via finding the upper bound and the lower bound for the blow-up times. (English) |
| Keyword:
|
nonlinear pseudoparabolic equation |
| Keyword:
|
Faedo-Galerkin method |
| Keyword:
|
local existence |
| Keyword:
|
blow-up |
| Keyword:
|
lifespan |
| Keyword:
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the global existence and decay of weak solutions |
| MSC:
|
35A01 |
| MSC:
|
35B40 |
| MSC:
|
35B44 |
| MSC:
|
35K70 |
| DOI:
|
10.21136/MB.2025.0059-24 |
| . |
| Date available:
|
2026-02-19T14:29:55Z |
| Last updated:
|
2026-02-19 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153389 |
| . |
| Reference:
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[1] Al'shin, A. B., Korpusov, M. O., Sveshnikov, A. G.: Blow-up in Nonlinear Sobolev Type Equations.De Gruyter Series in Nonlinear Analysis and Applications 15. Walter de Gruyter, Berlin (2011). Zbl 1259.35002, MR 2814745, 10.1515/9783110255294 |
| Reference:
|
[2] Amick, C. J., Bona, J. L., Schonbek, M. E.: Decay of solutions of some nonlinear wave equations.J. Differ. Equations 81 (1989), 1-49. Zbl 0689.35081, MR 1012198, 10.1016/0022-0396(89)90176-9 |
| Reference:
|
[3] Bona, J. L., Dougalis, V. A.: An initial- and boundary-value problem for a model equation for propagation of long waves.J. Math. Anal. Appl. 75 (1980), 503-522. Zbl 0444.35069, MR 0581837, 10.1016/0022-247X(80)90098-0 |
| Reference:
|
[4] Bouziani, A.: Solvability of nonlinear pseudoparabolic equation with a nonlocal boundary condition.Nonlinear Anal., Theory Methods Appl., Ser. A 55 (2003), 883-904. Zbl 1036.35109, MR 2017233, 10.1016/j.na.2003.07.011 |
| Reference:
|
[5] Bouziani, A.: Initial-boundary value problems for a class of pseudoparabolic equations with integral boundary conditions.J. Math. Anal. Appl. 291 (2004), 371-386. Zbl 1064.35095, MR 2038177, 10.1016/S0022-247X(03)00590-0 |
| Reference:
|
[6] Cao, Y., Wang, Z., Yin, J.: A note on the life span of semilinear pseudo-parabolic equation.Appl. Math. Lett. 98 (2019), 406-410. Zbl 1426.35041, MR 3980232, 10.1016/j.aml.2019.06.039 |
| Reference:
|
[7] Cao, Y., Yin, J., Wang, C.: Cauchy problems of semilinear pseudo-parabolic equations.J. Differ. Equations 246 (2009), 4568-4590. Zbl 1179.35178, MR 2523294, 10.1016/j.jde.2009.03.021 |
| Reference:
|
[8] Chen, S., Yu, J.: Dynamics of a diffusive predator-prey system with a nonlinear growth rate for the predator.J. Differ. Equations 260 (2016), 7923-7939. Zbl 1335.35114, MR 3479197, 10.1016/j.jde.2016.02.007 |
| Reference:
|
[9] Dai, D.-Q., Huang, Y.: A moment problem for one-dimensional nonlinear pseudoparabolic equation.J. Math. Anal. Appl. 328 (2007), 1057-1067. Zbl 1107.35072, MR 2290032, 10.1016/j.jmaa.2006.06.010 |
| Reference:
|
[10] Di, H., Shang, Y.: Blow-up of solutions for a class of nonlinear pseudoparabolic equations with a memory term.Abstr. Appl. Anal. 2014 (2014), Article ID 507494, 7 pages. Zbl 1470.35209, MR 3248862, 10.1155/2014/507494 |
| Reference:
|
[11] Di, H., Shang, Y.: Blow up of solutions for a class of fourth order nonlinear pseudo-parabolic equation with a nonlocal source.Bound. Value Probl. 2015 (2015), Article ID 109, 9 pages. Zbl 1338.35068, MR 3359758, 10.1186/s13661-015-0369-6 |
| Reference:
|
[12] Goudjo, C., Lèye, B., Sy, M.: Weak solution to a parabolic nonlinear system arising in biological dynamic in the soil.Int. J. Differ. Equ. 2011 (2011), Article ID 831436, 24 pages. Zbl 1235.35153, MR 2824911, 10.1155/2011/831436 |
| Reference:
|
[13] Kong, L., Wang, X., Zhao, X.: Asymptotic analysis to a parabolic system with weighted localized sources and inner absorptions.Arch. Math. 99 (2012), 375-386. Zbl 1258.35038, MR 2990156, 10.1007/s00013-012-0433-8 |
| Reference:
|
[14] Lèye, B., Doanh, N.-N., Monga, O., Garnier, P., Nunan, N.: Simulating biological dynamics using partial differential equations: Application to decomposition of organic matter in 3D soil structure.Vietnam J. Math. 43 (2015), 801-817. Zbl 1330.35469, MR 3422464, 10.1007/s10013-015-0159-6 |
| Reference:
|
[15] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non-linéaires.Études mathématiques. Dunod, Gauthier-Villars, Paris (1969), French. Zbl 0189.40603, MR 0259693 |
| Reference:
|
[16] Messaoudi, S. A., Talahmeh, A. A.: Blow up in a semilinear pseudo-parabolic equation with variable exponents.Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 65 (2019), 311-326. Zbl 1429.35036, MR 4026426, 10.1007/s11565-019-00326-1 |
| Reference:
|
[17] Meyvaci, M.: Blow up of solutions of pseudoparabolic equations.J. Math. Anal. Appl. 352 (2009), 629-633. Zbl 1160.35459, MR 2501907, 10.1016/j.jmaa.2008.11.016 |
| Reference:
|
[18] Ngoc, L. T. P., Long, N. T.: Corrigendum to: ``Existence, blow-up, and exponential decay estimates for a system of nonlinear wave equations with nonlinear boundary conditions''.Math. Methods Appl. Sci. 44 (2021), 7359-7368. Zbl 1471.35202, MR 4258427, 10.1002/mma.7250 |
| Reference:
|
[19] Ngoc, L. T. P., Nhan, N. H., Long, N. T.: General decay and blow-up results for a nonlinear pseudoparabolic equation with Robin-Dirichlet conditions.Math. Methods Appl. Sci. 44 (2021), 8697-8725. Zbl 1470.35068, MR 4279818, 10.1002/mma.7299 |
| Reference:
|
[20] Ngoc, L. T. P., Uyen, K. T. T., Nhan, N. H., Long, N. T.: On a system of nonlinear pseudoparabolic equations with Robin-Dirichlet boundary conditions.Commun. Pure Appl. Anal. 21 (2022), 585-623. Zbl 1483.35130, MR 4376310, 10.3934/cpaa.2021190 |
| Reference:
|
[21] Padrón, V.: Effect of aggregation on population recovery modeled by a forward-backward pseudoparabolic equation.Trans. Am. Math. Soc. 356 (2004), 2739-2756. Zbl 1056.35103, MR 2052595, 10.1090/S0002-9947-03-03340-3 |
| Reference:
|
[22] Popov, N. S.: Solvability of a boundary value problem for a pseudoparabolic equation with nonlocal integral conditions.Differ. Equ. 51 (2015), 362-375. Zbl 1331.35205, MR 3373207, 10.1134/S0012266115030076 |
| Reference:
|
[23] Shang, Y., Guo, B.: 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), 512-524 Chinese. Zbl 1057.45004, MR 2022221, 10.12387/C2003058 |
| Reference:
|
[24] Showalter, R. E.: Existence and representation theorems for a semilinear Sobolev equation in Banach space.SIAM J. Math. Anal. 3 (1972), 527-543. Zbl 0262.34047, MR 0315239, 10.1137/0503051 |
| Reference:
|
[25] 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, 10.58997/ejde.mon.01 |
| Reference:
|
[26] Showalter, R. E., Ting, T. W.: Pseudoparabolic partial differential equations.SIAM J. Math. Anal. 1 (1970), 1-26. Zbl 0199.42102, MR 0437936, 10.1137/0501001 |
| Reference:
|
[27] Showalter, R. E., Ting, T. W.: Asymptotic behavior of solutions of pseudo-parabolic partial differential equations.Ann. Mat. Pura Appl., IV. Ser. 90 (1971), 241-258. Zbl 0244.35012, MR 0316866, 10.1007/BF02415050 |
| Reference:
|
[28] Sobolev, S. L.: On a new problem of mathematical physics.Izv. Akad. Nauk SSSR, Ser. Mat. 18 (1954), 3-50 Russian. Zbl 0055.08401, MR 0069382 |
| Reference:
|
[29] Sun, F., Liu, L., Wu, Y.: Global existence and finite time blow-up of solutions for the semilinear pseudo-parabolic equation with a memory term.Appl. Anal. 98 (2019), 735-755. Zbl 1407.35122, MR 3908983, 10.1080/00036811.2017.1400536 |
| Reference:
|
[30] Tian, Y., Xiang, Z.: Global solutions to a 3D chemotaxis-Stokes system with nonlinear cell diffusion and Robin signal boundary condition.J. Differ. Equations 269 (2020), 2012-2056. Zbl 1439.35249, MR 4093726, 10.1016/j.jde.2020.01.031 |
| Reference:
|
[31] Ting, T. W.: Certain non-steady flows of second-order fluids.Arch. Ration. Mech. Anal. 14 (1963), 1-26. Zbl 0139.20105, MR 0153255, 10.1007/BF00250690 |
| Reference:
|
[32] Ting, T. W.: Parabolic and pseudo-parabolic partial differential equations.J. Math. Soc. Japan 21 (1969), 440-453. Zbl 0177.36701, MR 0264231, 10.2969/jmsj/02130440 |
| Reference:
|
[33] Vitillaro, E.: Global nonexistence theorems for a class of evolution equations with dissipation.Arch. Ration. Mech. Anal. 149 (1999), 155-182. Zbl 0934.35101, MR 1719145, 10.1007/s002050050171 |
| Reference:
|
[34] Xu, R., Su, J.: Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations.J. Funct. Anal. 264 (2013), 2732-2763. Zbl 1279.35065, MR 3045640, 10.1016/j.jfa.2013.03.010 |
| Reference:
|
[35] Yushkov, E. V.: Existence and blow-up of solutions of a pseudoparabolic equation.Differ. Equ. 47 (2011), 291-295. Zbl 1259.35126, MR 2895685, 10.1134/S0012266111020169 |
| Reference:
|
[36] Zennir, K., Miyasita, T.: Lifespan of solutions for a class of pseudo-parabolic equation with weak-memory.Alexandria Eng. J. 59 (2020), 957-964. 10.1016/j.aej.2020.03.016 |
| Reference:
|
[37] Zhang, L.: Decay of solution of generalized Benjamin-Bona-Mahony-Burgers equations in $n$-space dimensions.Nonlinear Anal., Theory Methods Appl. 25 (1995), 1343-1369. Zbl 0838.35118, MR 1355727, 10.1016/0362-546X(94)00252-D |
| Reference:
|
[38] Zhu, X., Li, F., Li, Y.: Global solutions and blow up solutions to a class pseudo-parabolic equations with nonlocal term.Appl. Math. Comput. 329 (2018), 38-51. Zbl 1427.35148, MR 3773309, 10.1016/j.amc.2018.02.003 |
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