Global existence and stability of solution for a p-Kirchhoff type hyperboc equation with variable exponents
DOI:
https://doi.org/10.5269/bspm.51464Resumen
In this paper, we consider the following p-Kirchhoff type hyperboc equation with variable exponents
Equation
We prove that a global existence of the solution with positive initial energy, the stability based of Komornik’s inequality.
Referencias
1. A. Benaissa, SA. Messaoudi, Blow-up of solutions for Kirchhoff equation of q-Laplacian type with nonlinear dissipation. Colloq. Math. 94(1), 103-109 (2002). https://doi.org/10.4064/cm94-1-8
2. H. Chen, GW. Liu, Global existence, uniform decay and exponential growth for a class of semilinear wave equation with strong damping. Acta Math. Sci. 33B(1), 41-58 (2013). https://doi.org/10.1016/S0252-9602(12)60193-3
3. Q. Gao, Y. Wang, Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation. Cent. Eur. J. Math. 9(3), 686-698 (2011). https://doi.org/10.2478/s11533-010-0096-2
4. V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source term, J Differ Equations 1994; 109: 295-308. https://doi.org/10.1006/jdeq.1994.1051
5. S. Ghegal, I. Hamchi and SA. Messaoudi, Global existence and stability of a nonlinear wave equation with variableexponent nonlinearities, Applicable Analysis, https://doi.org/10.1080/00036811.2018.1530760
6. Kirchhoff G, Mechanik, Teubner, 1883.
7. Komornik V, Exact controllability and stabilization the multiplier method. Paris: Masson-JohnWiley; 1994.
8. D. Lars, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, in: Lecture Notes in Mathematics, Vol. 2017, 2017.
9. HA. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form, Trans Amer Math Soc 1974; 192: 1-21. https://doi.org/10.1090/S0002-9947-1974-0344697-2
10. HA. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J Math Anal 1974; 5:138-146. https://doi.org/10.1137/0505015
11. S. Messaoudi, A. Talahmeh, H. Jamal, Nonlinear damped wave equation: existence and blow-up. Comput Math Appl. 2017; 74:3024-3041. https://doi.org/10.1016/j.camwa.2017.07.048
12. S. Messaoudi, A. Talahmeh, A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities. Appl Anal. 2017;96:1509-1515. https://doi.org/10.1080/00036811.2016.1276170
13. S. Messaoudi, A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities. Math Methods Appl Sci. 2017;40:1099-1476. https://doi.org/10.1002/mma.4505
14. K. Ono, Global existence, decay, and blow-up of solutions for some mildly degenerate nonlinear Kirchhoff strings, J Differ Equations 1997; 137: 273-301. https://doi.org/10.1006/jdeq.1997.3263
15. A. Ouaoua, M. Maouni, Blow-up, exponential growth of solution for a nonlinear parabolic equation with p(x)-Laplacian, International Journal of Analysis and Applications, V 17, N 4, (2019), 620-629.
16. A. Stanislav, S. Sergey, Evolution PDEs with nonstandard growth conditions: existence, uniqueness, localization, blowup. Atlantis Stud Differential Equations. 2015;4:1-417.
17. E. Vitillaro, Global existence theorems for a class of evolution equations with dissipation, Arch Rational Mech Anal 1999; 149: 155-182. https://doi.org/10.1007/s002050050171
18. ST. Wu, LY. Tsai, Blow-up solutions for some nonlinear wave equations of Kirchhoff type with some dissipation, Nonlinear Anal 2006; 65: 243-264. https://doi.org/10.1016/j.na.2004.11.023
19. YZ. Xu, Y. Ding, Global solutions and finite time blow-up for damped Klein-Gordon equation. Acta Math. Sci. 33B(1), 643-652 (2013). https://doi.org/10.1016/S0252-9602(13)60027-2
20. SQ. Yu, On the strongly damped wave equation with nonlinear damping and source terms. Electron. J. Qual. Theory Differ. Equ. 2009, 39 (2009).
2. H. Chen, GW. Liu, Global existence, uniform decay and exponential growth for a class of semilinear wave equation with strong damping. Acta Math. Sci. 33B(1), 41-58 (2013). https://doi.org/10.1016/S0252-9602(12)60193-3
3. Q. Gao, Y. Wang, Blow-up of the solution for higher-order Kirchhoff-type equations with nonlinear dissipation. Cent. Eur. J. Math. 9(3), 686-698 (2011). https://doi.org/10.2478/s11533-010-0096-2
4. V. Georgiev, G. Todorova, Existence of a solution of the wave equation with nonlinear damping and source term, J Differ Equations 1994; 109: 295-308. https://doi.org/10.1006/jdeq.1994.1051
5. S. Ghegal, I. Hamchi and SA. Messaoudi, Global existence and stability of a nonlinear wave equation with variableexponent nonlinearities, Applicable Analysis, https://doi.org/10.1080/00036811.2018.1530760
6. Kirchhoff G, Mechanik, Teubner, 1883.
7. Komornik V, Exact controllability and stabilization the multiplier method. Paris: Masson-JohnWiley; 1994.
8. D. Lars, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, in: Lecture Notes in Mathematics, Vol. 2017, 2017.
9. HA. Levine, Instability and nonexistence of global solutions to nonlinear wave equations of the form, Trans Amer Math Soc 1974; 192: 1-21. https://doi.org/10.1090/S0002-9947-1974-0344697-2
10. HA. Levine, Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J Math Anal 1974; 5:138-146. https://doi.org/10.1137/0505015
11. S. Messaoudi, A. Talahmeh, H. Jamal, Nonlinear damped wave equation: existence and blow-up. Comput Math Appl. 2017; 74:3024-3041. https://doi.org/10.1016/j.camwa.2017.07.048
12. S. Messaoudi, A. Talahmeh, A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities. Appl Anal. 2017;96:1509-1515. https://doi.org/10.1080/00036811.2016.1276170
13. S. Messaoudi, A. Talahmeh, Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities. Math Methods Appl Sci. 2017;40:1099-1476. https://doi.org/10.1002/mma.4505
14. K. Ono, Global existence, decay, and blow-up of solutions for some mildly degenerate nonlinear Kirchhoff strings, J Differ Equations 1997; 137: 273-301. https://doi.org/10.1006/jdeq.1997.3263
15. A. Ouaoua, M. Maouni, Blow-up, exponential growth of solution for a nonlinear parabolic equation with p(x)-Laplacian, International Journal of Analysis and Applications, V 17, N 4, (2019), 620-629.
16. A. Stanislav, S. Sergey, Evolution PDEs with nonstandard growth conditions: existence, uniqueness, localization, blowup. Atlantis Stud Differential Equations. 2015;4:1-417.
17. E. Vitillaro, Global existence theorems for a class of evolution equations with dissipation, Arch Rational Mech Anal 1999; 149: 155-182. https://doi.org/10.1007/s002050050171
18. ST. Wu, LY. Tsai, Blow-up solutions for some nonlinear wave equations of Kirchhoff type with some dissipation, Nonlinear Anal 2006; 65: 243-264. https://doi.org/10.1016/j.na.2004.11.023
19. YZ. Xu, Y. Ding, Global solutions and finite time blow-up for damped Klein-Gordon equation. Acta Math. Sci. 33B(1), 643-652 (2013). https://doi.org/10.1016/S0252-9602(13)60027-2
20. SQ. Yu, On the strongly damped wave equation with nonlinear damping and source terms. Electron. J. Qual. Theory Differ. Equ. 2009, 39 (2009).
Descargas
Publicado
2022-02-05
Número
Sección
Proceedings
Licencia
When the manuscript is accepted for publication, the authors agree automatically to transfer the copyright to the (SPM).
The journal utilize the Creative Common Attribution (CC-BY 4.0).
