Global existence and stability of solution for a p-Kirchhoff type hyperboc equation with variable exponents
Résumé
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 Komorniks inequality.
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Références
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
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
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
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
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
Kirchhoff G, Mechanik, Teubner, 1883.
Komornik V, Exact controllability and stabilization the multiplier method. Paris: Masson-JohnWiley; 1994.
D. Lars, P. Harjulehto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, in: Lecture Notes in Mathematics, Vol. 2017, 2017.
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
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
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
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
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
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
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.
A. Stanislav, S. Sergey, Evolution PDEs with nonstandard growth conditions: existence, uniqueness, localization, blowup. Atlantis Stud Differential Equations. 2015;4:1-417.
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
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
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
SQ. Yu, On the strongly damped wave equation with nonlinear damping and source terms. Electron. J. Qual. Theory Differ. Equ. 2009, 39 (2009).
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