Global existence and stability of a nonlinear wave equation with variable-exponent nonlinearities

References

• Ball J. Remarks on blow-up and nonexistence theorems for nonlinear evolution equations. Quart J Math. 1977;28:473–486. doi: 10.1093/qmath/28.4.473
   Web of Science ®Google Scholar
• Haraux A, Zuazua E. Decay estimates for some semilinear damped hyperbolic problems. Arch Ration Mech Anal. 1988;150:191–206. doi: 10.1007/BF00282203
   Web of Science ®Google Scholar
• Levine H. Some additional remarks on the nonexistence of global solutions to nonlinear wave equations. SIAM J Math Anal. 1974;5:138–146. doi: 10.1137/0505015
   Web of Science ®Google Scholar
• Georgiev V, Todorova G. Existence of a solution of the wave equation with nonlinear damping and source terms. J Differential Equations. 1994;109:295–308. doi: 10.1006/jdeq.1994.1051
   Web of Science ®Google Scholar
• Messaoudi S. Blow up in a nonlinearly damped wave equation. Math Nachr. 2001;231:1–7. doi: 10.1002/1522-2616(200111)231:1<105::AID-MANA105>3.0.CO;2-I
   Web of Science ®Google Scholar
• Ikehata R. Some remarks on the wave equations with nonlinear damping and source terms. Nonlinear Anal. 1995;27:1165–1175. doi: 10.1016/0362-546X(95)00119-G
   Web of Science ®Google Scholar
• Sattinger D. On global solution of nonlinear hyperbolic equations. Arch Ration Mech Anal. 1968;30:148–172. doi: 10.1007/BF00250942
   Web of Science ®Google Scholar
• Ha T. Blow-up for semilinear wave equation with boundary damping and source terms. J Math Anal Appl. 2012;390:328–334. doi: 10.1016/j.jmaa.2012.01.037
   Web of Science ®Google Scholar
• Jianghao H, Cai L. Global existence and blowup of solutions for nonlinear coupled wave equations with viscoelastic terms. Math Methods Appl Sci. 2016;39:1977–1989. doi: 10.1002/mma.3614
   Web of Science ®Google Scholar
• Jianghao H, Yajing Z, Shengjia L. Global existence and blow-up phenomena for a nonlinear wave equation. Nonlinear Anal. 2009;71:4823–4832. doi: 10.1016/j.na.2009.03.058
   Web of Science ®Google Scholar
• Park J, Ha T. Existence and asymptotic stability for the semilinear wave equation with boundary damping and source term. J Math Phys. 2008;49:053511. doi: 10.1063/1.2919886
   Web of Science ®Google Scholar
• Diening L, Hasto P, Harjulehto P, et al. Lebesgue and Sobolev spaces with variable exponents. Heidelberg: Springer-Verlag; 2011. (Springer Lecture Notes; 2017).
   Google Scholar
• Stanislav A, Sergey S. Evolution PDEs with nonstandard growth conditions: existence, uniqueness, localization, blow-up. Atlantis Stud Differential Equations. 2015;4:1–417. doi: 10.2991/978-94-6239-112-3_1
   Google Scholar
• Antontsev S. Wave equation with p(x, t)-Laplacian and damping term: existence and blow-up. J Difference Equ Appl. 2011;3:503–525.
   Google Scholar
• Gao Y, Gao W. Existence of solutions for nonlocal p-Laplacian thermistor problems on time scales. Bound Value Probl. 2013;2013:1–8. doi: 10.1186/1687-2770-2013-1
   Web of Science ®Google Scholar
• Sun L, Ren Y, Gao W. Lower and upper bounds for the blow-up time for nonlinear wave equation with variable sources. Comput Math Appl. 2016;71:267–277. doi: 10.1016/j.camwa.2015.11.016
   Web of Science ®Google Scholar
• Messaoudi S, Talahmeh A. A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities. Appl Anal. 2017;96:1509–1515. doi: 10.1080/00036811.2016.1276170
   Web of Science ®Google Scholar
• Messaoudi S, Talahmeh A. Blowup in solutions of a quasilinear wave equation with variable-exponent nonlinearities. Math Methods Appl Sci. 2017;40:1099–1476. doi: 10.1002/mma.4505
   Web of Science ®Google Scholar
• Messaoudi S, Talahmeh A, Jamal H. Nonlinear damped wave equation: existence and blow-up. Comput Math Appl. 2017;74:3024–3041. doi: 10.1016/j.camwa.2017.07.048
   Web of Science ®Google Scholar
• Aboulaich R, Meskine D, Souissi A. New diffusion models in image processing. Comput Math Appl. 2008;56:874–882. doi: 10.1016/j.camwa.2008.01.017
   Web of Science ®Google Scholar
• Antontsev S, Zhikov V. Higher integrability for parabolic equation of p(x, t)-Laplacian type. Adv Differential Equation. 2005;10:1053–1080.
   Web of Science ®Google Scholar
• Antontsev S, Shmarev S. Elliptic equations with anisotropic nonlinearity and nonstandard growth condition. In: Chipot M, Quittner P, editors. Handbook of differential equations. Stationary partial differential equations. Vol. 3. Amsterdam: Elsevier/North Holland; 2006. p. 1–100.
   Google Scholar
• Antontsev S, Shmarev S. Blow-up of solutions to parabolic equations with nonstandard growth conditions. J Comput Appl Math. 2010;234:2633–2645. doi: 10.1016/j.cam.2010.01.026
   Web of Science ®Google Scholar
• Komornik V. Exact controllability and stabilization the multiplier method. Paris: Masson-John Wiley; 1994.
   Google Scholar