| Title:
|
Global existence and stability of solution for a nonlinear Kirchhoff type reaction-diffusion equation with variable exponents (English) |
| Author:
|
Khaldi, Aya |
| Author:
|
Ouaoua, Amar |
| Author:
|
Maouni, Messaoud |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
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2464-7136 (online) |
| Volume:
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147 |
| Issue:
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4 |
| Year:
|
2022 |
| Pages:
|
471-484 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We consider a class of Kirchhoff type reaction-diffusion equations with variable exponents and source terms \begin {equation*} u_{t}-M\biggl (\int _\Omega \vert \nabla u \vert ^{2} {\rm d}x\bigg ) \Delta u+ \vert u \vert ^{m(x) -2}u_{t}= \vert u \vert ^{r(x) -2}u. \end {equation*} We prove with suitable assumptions on the variable exponents $r( {\cdot }),$ $m({\cdot })$ the global existence of the solution and a stability result using potential and Nihari's functionals with small positive initial energy, the stability being based on Komornik's inequality. (English) |
| Keyword:
|
Kirchhoff equation |
| Keyword:
|
reaction-diffusion equation |
| Keyword:
|
variable exponent |
| Keyword:
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global solution |
| MSC:
|
35B40 |
| MSC:
|
35L10 |
| MSC:
|
35L70 |
| idZBL:
|
Zbl 07655821 |
| idMR:
|
MR4512168 |
| DOI:
|
10.21136/MB.2021.0122-20 |
| . |
| Date available:
|
2022-11-16T11:16:11Z |
| Last updated:
|
2023-04-11 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151093 |
| . |
| Reference:
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[1] Antontsev, S., Shmarev, S.: Evolution PDEs with Nonstandard Growth Conditions: Existence, Uniqueness, Localization, Blow-up.Atlantis Studies in Differential Equations 4. Springer, Berlin (2015). Zbl 1410.35001, MR 3328376, 10.2991/978-94-6239-112-3 |
| Reference:
|
[2] Benaissa, A., Messaoudi, S. A.: Blow-up of solutions for Kirchhoff equation of $q$-Laplacian type with nonlinear dissipation.Colloq. Math. 94 (2002), 103-109. Zbl 1090.35122, MR 1930205, 10.4064/cm94-1-8 |
| Reference:
|
[3] Chen, H., Liu, G.: Global existence, uniform decay and exponential growth for a class of semi-linear wave equations with strong damping.Acta Math. Sci., Ser. B, Engl. Ed. 33 (2013), 41-58. Zbl 1289.35202, MR 3003742, 10.1016/s0252-9602(12)60193-3 |
| Reference:
|
[4] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents.Lecture Notes in Mathematics 2017. Springer, Berlin (2011). Zbl 1222.46002, MR 2790542, 10.1007/978-3-642-18363-8 |
| Reference:
|
[5] Fu, Y., Xiang, M.: Existence of solutions for parabolic equations of Kirchhoff type involving variable exponent.Appl. Anal. 95 (2016), 524-544. Zbl 1334.35079, MR 3440345, 10.1080/00036811.2015.1022153 |
| Reference:
|
[6] Gao, Q., Li, F., Wang, Y.: Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation.Cent. Eur. J. Math. 9 (2011), 686-698. Zbl 1233.35145, MR 2784038, 10.2478/s11533-010-0096-2 |
| Reference:
|
[7] Ghegal, S., Hamchi, I., Messaoudi, S. A.: Global existence and stability of a nonlinear wave equation with variable-exponent nonlinearities.Appl. Anal. 99 (2020), 1333-1343. Zbl 1439.35333, MR 4097821, 10.1080/00036811.2018.1530760 |
| Reference:
|
[8] Ghisi, M., Gobbino, M.: Hyperbolic-parabolic singular perturbation for middly degenerate Kirchhoff equations: Time-decay estimates.J. Differ. Equations 245 (2008), 2979-3007. Zbl 1162.35008, MR 2454809, 10.1016/j.jde.2008.04.017 |
| Reference:
|
[9] Han, Y., Li, Q.: Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy.Comput. Math. Appl. 75 (2018), 3283-3297. Zbl 1409.35143, MR 3785559, 10.1016/j.camwa.2018.01.047 |
| Reference:
|
[10] Jiang, Z., Zheng, S., Song, X.: Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions.Appl. Math. Lett. 17 (2004), 193-199. Zbl 1056.35087, MR 2034767, 10.1016/S0893-9659(04)90032-8 |
| Reference:
|
[11] Kirchhoff, G.: Vorlesungen über mathematische Physik. 1. Band: Mechanik.Teubner, Leipzig (1883), German. |
| Reference:
|
[12] Komornik, V.: Exact Controllability and Stabilization: The Multiplier Method.Research in Applied Mathematics 36. Wiley, Chichester (1994). Zbl 0937.93003, MR 1359765 |
| Reference:
|
[13] Levine, H. A.: Instability and nonexistence of global solutions to nonlinear wave equations of the form $Pu_{tt}=-Au+{\cal F}(u)$.Trans. Am. Math. Soc. 192 (1974), 1-21. Zbl 0288.35003, MR 0344697, 10.1090/S0002-9947-1974-0344697-2 |
| Reference:
|
[14] Levine, H. A.: Some additional remarks on the nonexistence of global solutions to nonlinear wave equations.SIAM J. Math. Anal. 5 (1974), 138-146. Zbl 0243.35069, MR 0399682, 10.1137/0505015 |
| Reference:
|
[15] Li, H.: Blow-up of solutions to a $p$-Kirchhoff-type parabolic equation with general nonlinearity.J. Dyn. Control Syst. 26 (2020), 383-392. Zbl 1445.35206, MR 4068420, 10.1007/s10883-019-09463-4 |
| Reference:
|
[16] Li, J., Han, Y.: Global existence and finite time blow-up of solutions to a nonlocal $p$-Laplace equation.Math. Model. Anal. 24 (2019), 195-217. Zbl 07394651, MR 3917481, 10.3846/mma.2019.014 |
| Reference:
|
[17] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires.Gauthier-Villars, Paris (1969), French. Zbl 0189.40603, MR 0259693 |
| Reference:
|
[18] Messaoudi, S. A., Talahmeh, A. A.: Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities.Math. Methods Appl. Sci. 40 (2017), 6976-6986. Zbl 1397.35042, MR 3742108, 10.1002/mma.4505 |
| Reference:
|
[19] 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:
|
[20] Messaoudi, S. A., Talahmeh, A. A., Al-Smail, J. H.: Nonlinear damped wave equation: Existence and blow-up.Comput. Math. Appl. 74 (2017), 3024-3041. Zbl 1415.35061, MR 3725935, 10.1016/j.camwa.2017.07.048 |
| Reference:
|
[21] Ono, K.: Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings.J. Differ. Equations 137 (1997), 273-301. Zbl 0879.35110, MR 1456598, 10.1006/jdeq.1997.3263 |
| Reference:
|
[22] Ouaoua, A., Maouni, M.: Blow-up, exponential growth of solution for a nonlinear parabolic equation with $p(x)$-Laplacian.Int. J. Anal. Appl. 17 (2019), 620-629. Zbl 1438.35226, 10.28924/2291-8639-17-2019-620 |
| Reference:
|
[23] Polat, N.: Blow up of solution for a nonlinear reaction diffusion equation with multiple nonlinearities.Int. J. Sci. Technol. 2 (2007), 123-128. MR 2372361 |
| Reference:
|
[24] 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:
|
[25] Wu, S. T., Tsai, L.-Y.: Blow-up solutions for some non-linear wave equations of Kirchhoff type with some dissipation.Nonlinear Anal., Theory Methods Appl., Ser. A 65 (2006), 243-264. Zbl 1151.35052, MR 2228427, 10.1016/j.na.2004.11.023 |
| Reference:
|
[26] Zheng, S.: Nonlinear Evolution Equations.Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics 133. Chapman & Hall/CRC, Boca Raton (2004). Zbl 1085.47058, MR 2088362, 10.1201/9780203492222 |
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