×

Decay and blow-up for a viscoelastic wave equation of variable coefficients with logarithmic nonlinearity. (English) Zbl 1475.35044

Summary: In this article, we consider a viscoelastic wave equation of variable coefficients with logarithmic nonlinearity and dynamic boundary conditions in a bounded domain. The existence of a global solution is given by use of the potential well method. In the stable set, with some suitable assumptions, we establish an explicit and general decay rate result without imposing restrictive assumptions on the behavior of the relaxation function at infinity by use of the Riemannian geometry method, logarithmic Sobolev inequality, and Lyapunov functional method. Meanwhile, in the unstable set, blow-up of the solution is also obtained.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B44 Blow-up in context of PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35L71 Second-order semilinear hyperbolic equations
35R09 Integro-partial differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Al-Gharabli, M. M.; Guesmia, A.; Messaoudi, S. A., Existence and a general decay results for a viscoelastic plate equation with a logarithmic nonlinearity, Commun. Pure Appl. Anal., 18, 159-180 (2019) · Zbl 1400.35177
[2] Barrow, J. D.; Parsons, P., Inflationary models with logarithmic potentials, Phys. Rev. D, 52, 5576-5587 (1995)
[3] Bartkowski, K.; Przemysaw, Gorka, One-dimensional Klein-Gordon equation with logarithmic nonlinearities, J. Phys. A, Math. Theor., 41, Article 355201 pp. (2008) · Zbl 1146.81021
[4] Bouhali, K.; Ellaggoune, F., Viscoelastic wave equation with logarithmic nonlinearities in \(R^n\), J. Partial Differ. Equ., 30, 47-63 (2017) · Zbl 1389.35216
[5] Dafermos, C. M., Asymptotic stability in viscoelasticity, Arch. Ration. Mech. Anal., 37, 297-308 (1970) · Zbl 0214.24503
[6] Enqvist, K.; Mcdonald, J., Q-balls and baryogenesis in the MSSM, Phys. Lett. B, 425, 309-321 (1998)
[7] Feng, B. W., General decay rates for a viscoelastic wave equation with dynamic boundary conditions and past history, Mediterr. J. Math., 15, 1-17 (2018) · Zbl 1403.35043
[8] Gazzola, F.; Squassina, M., Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. Henri PoincarĂ©, 23, 185-207 (2006) · Zbl 1094.35082
[9] Gorka, P., Logarithmic Klein-Gordon equation, Acta Phys. Pol. B, 40, 59-66 (2009) · Zbl 1371.81101
[10] Gross, L., Logarithmic Sobolev inequalities, Am. J. Math., 97, 1061-1083 (1975) · Zbl 0318.46049
[11] Guesmia, A., Asymptotic stability of abstract dissipative systems with infinite memory, J. Math. Anal., 382, 748-760 (2011) · Zbl 1225.45005
[12] Guesmia, A., Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control Inf., 30, 507-526 (2013) · Zbl 1279.93090
[13] Guesmia, A.; Messaoudi, S. A., A new approach to the stability of an abstract system in the presence of infinite history, J. Math. Anal. Appl., 416, 212-228 (2014) · Zbl 1304.35091
[14] Guo, B. Z.; Shao, Z. C., On exponential stability of a semilinear wave equation with variable coefficients under the nonlinear boundary feedback, Nonlinear Anal., 71, 5961-5978 (2009) · Zbl 1194.35269
[15] Hamchi, I., Uniform decay rates for second-order hyperbolic equations with variable coefficients, Asymptot. Anal., 57, 71-82 (2008) · Zbl 1148.35303
[16] Han, X., Global existence of weak solutions for a logarithmic wave equation arising from Q-ball dynamics, Bull. Korean Math. Soc., 50, 275-283 (2013) · Zbl 1262.35150
[17] Hao, J. H.; Du, F. Q., Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions, Math. Methods Appl. Sci., 44, 284-302 (2021) · Zbl 1469.35034
[18] Hao, J. H.; Lv, M. X., Energy decay for variable coefficient viscoelastic wave equation with acoustic boundary conditions in domains with nonlocally reacting boundary, Electron. J. Differ. Equ., 95, Article 1 pp. (2020)
[19] Hao, J. H.; Wang, P. P., Uniform stability of transmission of wave-plate equations with source on Riemannian manifold, J. Differ. Equ., 268, 6385-6415 (2020) · Zbl 1435.35238
[20] Hiramatsu, T.; Kawasaki, M.; Takahashi, F., Numerical study of Q-ball formation in gravity mediation, J. Cosmol. Astropart. Phys., 2010, 6, Article 008 pp. (2010)
[21] Hu, B.; Yin, H. M., Semilinear parabolic equations with prescribed energy, Rend. Circ. Mat. Palermo, 44, 479-505 (1995) · Zbl 0856.35063
[22] Krolikowski, W.; Edmundson, D.; Bang, O., Unified model for partially coherent solitons in logarithmically nonlinear media, Phys. Rev. E, 61, 3122-3126 (2000)
[23] Liu, Y. X.; Li, J.; Yao, P. F., Decay rates of the hyperbolic equation in an exterior domain with half-linear and nonlinear boundary dissipations, J. Syst. Sci. Complex., 29, 657-680 (2016) · Zbl 1346.93199
[24] Marcelo, M.; Cavalcanti, Khemmoudj A.; Medjden, M., Uniform stabilization of the damped Cauchy-Ventcel problem with variable coefficients and dynamic boundary conditions, J. Math. Anal. Appl., 328, 900-930 (2007) · Zbl 1107.35024
[25] Messaoudi, S. A., Global existence and nonexistence in a system of Petrovsky, J. Math. Anal. Appl., 265, 296-308 (2002) · Zbl 1006.35070
[26] Nehari, Z., On a class of nonlinear second-order differential equations, Trans. Am. Math. Soc., 95, 101-123 (1960) · Zbl 0097.29501
[27] Payne, L. E.; Sattinger, D. H., Saddle points and instability of nonlinear hyperbolic equations, Isr. J. Math., 22, 273-303 (1975) · Zbl 0317.35059
[28] Wang, P. P.; Hao, J. H., Viscoelastic versus frictional dissipation in a variable coefficients plate system with time-varying delay, Z. Angew. Math. Phys., 70, 1-19 (2019) · Zbl 1428.35582
[29] Yao, P. F., On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim., 37, 1568-1599 (1999) · Zbl 0951.35069
[30] Yao, P. F., Global smooth solutions for the quasilinear wave equation with boundary dissipation, J. Differ. Equ., 241, 62-93 (2007) · Zbl 1214.35037
[31] Yao, P. F., Boundary controllability for the quasilinear wave equation, Appl. Math. Optim., 61, 191-233 (2010) · Zbl 1185.93018
[32] Ye, Y. J., Global solution and blow-up of logarithmic Klein-Gordon equation, Bull. Korean Math. Soc., 57, 281-294 (2020) · Zbl 1445.35088
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.