Global Existence, Stability and Blow–up of Solutions for p-Biharmonic Hyperbolic Equation with Weak and Strong Damping Terms
Анотація
У цій статті ми досліджуємо початково-крайову задачу для $p\text{-}$бігармонічного гіперболічного рівняння зі слабкими та сильними демпфувальними членами: $$ v_{tt}+\Delta_{p}^{2}v-\mu\Delta_{m}v_{t}+v_{t}=\omega|v|^{k-2}v.$$ При деяких припущеннях на початкові дані, сталі $p$, $m$ та $k$, ми довели глобальне існування, стійкість та результати стосовно руйнування розв'язків. Глобальний розв'язок одержано методом потенціальної ями, а стійкість ґрунтується на нерівності Коморніка. Також доведено, що розв'язок з від'ємною початковою енергією вибухає за скінченний та за нескінченний час.
Mathematical Subject Classification 2020: 35L75, 35A01, 35B35
Ключові слова:
$p$-бігармонічне рівняння, демпфувальні члени, глобальне існування, стійкість, руйнуванняПосилання
N. Boumaza and B. Gheraibia, On the existence of a local solution for an integro-differential equation with an integral boundary condition, Bol. Soc. Mat. Mex. 26 (2020), 521--534. https://doi.org/10.1007/s40590-019-00266-y
N. Boumaza and B. Gheraibia, and G. Liu, Global well-posedness of solutions for the $p$-laplacian hyperbolic type equation with weak and strong damping terms and logarithmic nonlinearity, Taiwanese J. Math. 26 (2022), 1235--1255. https://doi.org/10.11650/tjm/220702
P. Candito and R. Livrea, Infinitely many solutions for a nonlinear Navier boundary value problem involving the $p$-biharmonic, Studia Univ. Babec{s}-Bolyai Math. LV (2010), 41--51.
W. Chen and Y. Zhou, Global nonexistence for a semilinear Petrovsky equation, Nonlinear Anal. 70 (2009), 3203--3208. https://doi.org/10.1016/j.na.2008.04.024
L. Ding, Multiple solutions for a perturbed Navier boundary value problem involving the $p$-biharmonic, Bull. Iranian Math. Soc. 41 (2015), 269--280.
P. Drabek and M. Otani, Global bifurcation result for the $p$-biharmonic operator, Electron. J. Differential Equations 2001 (2001), Paper No. 48, 19 pp.
J. Ferreira, W.S. Panni, S.A. Messaoudi, E. Pic{s}kin, and M. Shahrouzi, Existence and asymptotic behavior of beam-equation solutions with strong damping and $p(x)$-biharmonic operator, J. Math. Phys. Anal. Geom. 18 (2022), 488--513.
A. Hao and J. Zhou, Blowup, extinction and non-extinction for a nonlocal pbiharmonic parabolic equation, Appl. Math. Lett. 64 (2017), 198--204. https://doi.org/10.1016/j.aml.2016.09.007
Y. Huang and X. Liu, Sign-changing solutions for $p$-biharmonic equations with Hardy potential, J. Math. Anal. App. 412 (2014), 142--154. https://doi.org/10.1016/j.jmaa.2013.10.044
V. Komornik, Exact Controllability and Stabilization: the Multiplier Method, Masson-John Wiley, Paris, 1994.
A.C. Lazer and P.J. McKenna, Large amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis, SIAM Rev. 32 (1990), 537--578. https://doi.org/10.1137/1032120
C. Liu and J. Guo, Weak solutions for a fourth order degenerate parabolic equation, Bull. Pol. Acad. Sci. Math. 54 (2006), 27-39. https://doi.org/10.4064/ba54-1-3
G. Li, Y. Sun, and W. Liu, Global existence and blow up of solutions for a strongly damped Petrovsky system with nonlinear damping, Appl. Anal. 91 (2012), 575--586. https://doi.org/10.1080/00036811.2010.550576
H. Liu and H. Chen, Ground-state solution for a class of biharmonic equations including critical exponent, Z. Angew. Math. Phys. 66 (2015), 3333--3343. https://doi.org/10.1007/s00033-015-0583-1
X. Liu, H. Chen, and B. Almuaalemi, Ground state solutions for $p$-biharmonic equations, Electron. J. Differential Equations 2017(45) (2017), 1-9.
C. Liu and P. Li, A parabolic $p$-biharmonic equation with logarithmic nonlinearity, Politehn. Univ. Bucharest Sci. Bull. Ser. A Appl. Math. Phys. 81 (2019), 35--48.
Z. Liu and Z. Fang, Well-posedness and asymptotic behavior for a pseudo-parabolic equation involving $p$-biharmonic operator and logarithmic nonlinearity, Taiwanese J. Math. 27 (2023), 487--523. https://doi.org/10.11650/tjm/221106
P.J. McKenna and W. Walter, Nonlinear oscillations in a suspension bridge, Arch. Rational Mech. Anal. 98 (1987), 167--177. https://doi.org/10.1007/BF00251232
P.J. McKenna and W. Walter, Travelling waves in a suspension bridge, SIAM J. Appl. Math. 50 (1990), 703--715. https://doi.org/10.1137/0150041
S. Messaoudi, Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl. 265 (2002) 296--308. https://doi.org/10.1006/jmaa.2001.7697
G. Molica Bisci and D. Repovš, Multiple solutions of $p$-biharmonic equations with Navier boundary conditions, Compl. Vari. Ellip. Equ. 59 (2015), 271--284. https://doi.org/10.1080/17476933.2012.734301
D. Pereira, C.A. Raposo, and C.H.M. Maranhão, Global solution and asymptotic behaviour for a wave equation type $p$-Laplacian with $p$-Laplacian damping, MathLAB Journal 5 (2020), 35--45.
E. Pişkin and N. Polat, On the decay of solutions for a nonlinear Petrovsky equation, Mathematical Sciences Letters 3 (2014), 43--47. https://doi.org/10.12785/msl/030107
Y. Wang and Y. Shen, Multiple and sign-changing solutions for a class of semilinear biharmonic equation, J. Differential Equations. 246 (2009), 3109--3125. https://doi.org/10.1016/j.jde.2009.02.016
J. Wang and C. Liu, $p$-biharmonic parabolic equations with logarithmic nonlinearity, Electron. J. Differential Equations 2019 (2019), 1--18.
S.T. Wu and L.Y. Tsai, On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system, Taiwanese J. Math. 13(2A) (2009), 545--558. https://doi.org/10.11650/twjm/1500405355
J. Zhang, Existence results for some fourth-order nonlinear elliptic problems, Nonlinear Anal. 45 (2001), 29--36. https://doi.org/10.1016/S0362-546X(99)00328-4
