### Electronic Research Archive

2021, Issue 6: 3867-3887. doi: 10.3934/era.2021066

# A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms

• Received: 01 June 2021 Revised: 01 July 2021 Published: 07 September 2021
• Primary: 35A01, 35L75; Secondary: 35B40, 35B44

• In this paper, we study a class of hyperbolic equations of the fourth order with strong damping and logarithmic source terms. Firstly, we prove the local existence of the weak solution by using the contraction mapping principle. Secondly, in the potential well framework, the global existence of weak solutions and the energy decay estimate are obtained. Finally, we give the blow up result of the solution at a finite time under the subcritical initial energy.

Citation: Yi Cheng, Ying Chu. A class of fourth-order hyperbolic equations with strongly damped and nonlinear logarithmic terms[J]. Electronic Research Archive, 2021, 29(6): 3867-3887. doi: 10.3934/era.2021066

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• In this paper, we study a class of hyperbolic equations of the fourth order with strong damping and logarithmic source terms. Firstly, we prove the local existence of the weak solution by using the contraction mapping principle. Secondly, in the potential well framework, the global existence of weak solutions and the energy decay estimate are obtained. Finally, we give the blow up result of the solution at a finite time under the subcritical initial energy.

 [1] The existence and the asymptotic behavior of a plate equation with frictional damping and a logarithmic source term. J. Math. Anal. Appl. (2017) 454: 1114-1128. [2] Existence and a general decay result for a plate equation with nonlinear damping and a logarithmic source term. J. Evol. Equ. (2018) 18: 105-125. [3] An elliptic system with logarithmic nonlinearity. Adv. Nonlinear Anal. (2019) 8: 928-945. [4] Inflationary models with logarithmic potentials. Phys. Rev. D (1995) 52: 5576-5587. [5] K. Bartkowski and P. Górka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A, 41 (2008), 355201, 11 pp. doi: 10.1088/1751-8113/41/35/355201 [6] Wave equations with logarithmic nonlinearities. Bull. Acad. Polon. Sci. Ser. Sci. Math. Astronom. Phys. (1975) 23: 461-466. [7] A boundary value problem for fourth-order elastic beam equations. J. Math. Anal. Appl. (2008) 343: 1166-1176. [8] On the Sobolev space of functions with derivative of logarithmic order. Adv. Nonlinear Anal. (2020) 9: 836-849. [9] Équations d'évolution avec non-linéarité logarithmique. Ann. Fac. Sci. Toulouse Math. (1980) 2: 21-51. [10] On some fourth-order semilinear elliptic problems in $R^N$. Nonlinear Anal. (2002) 49: 861-884. [11] Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity. J. Differential Equations (2015) 258: 4424-4442. [12] Global existence and blow-up of solutions for infinitely degenerate semilinear pseudo-parabolic equations with logarithmic nonlinearity. Discrete Contin. Dyn. Syst. (2019) 39: 1185-1203. [13] Global nonexistence for a semilinear Petrovsky equation. Nonlinear Anal. (2009) 70: 3203-3208. [14] Existence and uniform decay estimates for the fourth order wave equation with nonlinear boundary damping and interior source. Electron. Res. Arch. (2020) 28: 221-261. [15] Q-balls and baryogenesis in the MSSM. Phys. Lett. B (1998) 425: 309-321. [16] L. C. Evans, Partial Differential Equations, Second ed., in: Graduate Studies in Mathematics, vol. 19, 2010. doi: 10.1090/gsm/019 [17] Global solutions and finite time blow up for damped semilinear wave equations. Ann. Inst. H. Poincaré Anal. Non Linéaire (2006) 23: 185-207. [18] Logarithmic Klein-Gordon equation. Acth Physica Polonica B (2009) 40: 59-66. [19] Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics. Bull. Korean Math. Soc. (2013) 50: 275-283. [20] Y. Han, Finite time blow up for a semilinear pseudo-parabolic equation with general nonlinearity, Appl. Math. Lett., 99 (2020), 105986, 7 pp. doi: 10.1016/j.aml.2019.07.017 [21] Numerical study of Q-ball formation in gravity mediation. J. Cosmol. Astropart. P. (2010) 6: 008-008. [22] Local existence and blow up of solutions to a logarithmic nonlinear wave equation with delay. Appl. Anal. (2020) 99: 530-547. [23] Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t = -Au+ F(u)$. Arch. Ration. Mech. Anal. (1973) 51: 371-386. [24] P. Li and C. Liu, A class of fourth-order parabolic equation with logarithmic nonlinearity, J. Inequal. Appl., (2018), Paper No. 328, 21 pp. doi: 10.1186/s13660-018-1920-7 [25] Global existence and blow up of solution for semilinear hyperbolic equation with logarithmic nonlinearity. Nonlinear Anal. (2019) 184: 239-257. [26] Global existence and blow up of solution for semi-linear hyperbolic equation with the product of logarithmic and power-type nonlinearity. Opuscula Math. (2020) 40: 111-130. [27] Global existence and blow up of solutions for pseudo-parabolic equation with singular potential. J. Differ. Equations (2020) 269: 4914-4959. [28] Global well-posedness of nonlinear wave equation with weak and strong damping terms and logarithmic source term. Adv. Nonlinear Anal. (2020) 9: 613-632. [29] Global existence and energy decay estimates for weak solutions to the pseudo-parabolic equation with variable exponents. Math. Method. Appl. Sci. (2020) 43: 2516-2527. [30] J.-L. Lions, Quelque Méthodes de Résolution des Problemes aux Limites non Linéaires, Dunod, Paris, 1969. [31] The existence, general decay and blow-up for a plate equation with nonlinear damping and a logarithmic source term. Electron. Res. Arch. (2020) 28: 263-289. [32] Initial-boundary value problem for a fourth-order plate equation with Hardy-Hénon potential and polynomial nonlinearity. Electron. Res. Arch. (2020) 28: 599-625. [33] Global existence and nonexistence in a system of petrovsky. J. Math. Anal. Appl. (2002) 265: 296-308. [34] Smooth attractors for strongly damped wave equations. Nonlinearity (2006) 19: 1495-1506. [35] Compact sets in the space $L^p(0, T;B)$. Ann. Mat. Pura Appl. (1987) 146: 65-96. [36] Pointwise gradient bounds for a class of very singular quasilinear elliptic equations. Discrete Contin. Dyn. Syst. (2021) 41: 4461-4476. [37] Global existence and finite time blowup for a nonlocal semilinear pseudo-parabolic equation. Adv. Nonlinear Anal. (2021) 10: 261-288. [38] On global solutions and blow-up of solutions for a nonlinearly damped Petrovsky system. Taiwan. J. Math. (2009) 13: 545-558. [39] Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations. J. Funct. Anal. (2013) 264: 2732-2763. [40] Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations. Opuscula Math. (2019) 39: 297-313. [41] On the logarithmic Keller-Segel-Fisher/KPP system. Discrete Contin. Dyn. Syst. (2019) 39: 5365-5402. [42] Exponential decay of energy for a logarithmic wave equation. J. Partial Differ. Equ. (2015) 28: 269-277. [43] X. Zhu, B. Guo and M. Liao, Global existence and blow-up of weak solutions for a pseudo-parabolic equation with high initial energy, Appl. Math. Lett., 104 (2020), 106270, 7 pp. doi: 10.1016/j.aml.2020.106270
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