BLOW UP OF SOLUTIONS FOR A NONLINEAR VISCOELASTIC WAVE EQUATIONS WITH VARIABLE EXPONENTS

Year 2019, Volume: 5 Issue: 2, 134 - 145, 30.12.2019

Erhan Pişkin

https://doi.org/10.23884/mejs.2019.5.2.05

Cited By: 14

Abstract

The aim of this work is to study the blow up of solutions for the viscoelastic wave equation with variable exponents in a bounded domain. Our result extends the one in <cite>Messaoudi1</cite> to problems with variable exponent nonlinearities.

Keywords

Blow up, Viscoelastic wave equation, Variable exponent

References

• Ball J. M., "Remarks on blow-up and nonexistence theorems for nonlinear evolution equations", Quart. J. Math. Oxford Ser., 28, 473-486, 1977.
• Cavalcanti M.M., Domingos Cavalcanti V.N., Ferreira J., "Existence and uniform decay for nonlinear viscoelastic equation with strong damping", Math. Methods Appl. Sci., 24, 1043-1053, 2001.
• Chen Y., Levine S., Rao M., "Variable Exponent, Linear Growth Functionals in Image Restoration", SIAM Journal on Applied Mathematics, 66, 1383-1406, 2006.
• Diening L., Hasto P., Harjulehto P., Ruzicka M.M., "Lebesgue and Sobolev Spaces with Variable Exponents", Springer-Verlag, 2011. Fan X.L., Shen J.S., Zhao D., "Sobolev embedding theorems for spaces W^{k,p(x)}(Ω)", J. Math. Anal. Appl., 263, 749-760, 2001.
• Georgiev V., Todorova G., "Existence of a solution of the wave equation with nonlinear damping and source term", J. Differ. Equations, 109, 295-308, 1994.
• Kalantarov V.K., Ladyzhenskaya O.A., "The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types", J. Soviet Math., 10, 53-70, 1978.
• Kovacik O., Rakosnik J., "On spaces L^{p(x)}(Ω), and W^{k,p(x)}(Ω)", Czechoslovak Mathematical Journal, 41, 592-618, 1991.
• Levine H.A., "Instability and nonexistence of global solutions of nonlinear wave equations of the form Pu_{tt}=Au+F(u)", Trans. Amer. Math. Soc., 192, 1-21, 1974.
• Messaoudi S.A., "Blow up and global existence in a nonlinear viscoelastic wave equation", Math. Nachr., 260 58-66, 2003.
• Messaoudi S.A., "Blow-up of positive-initial-energy solutions of a nonlinear viscoelastic hyperbolic equation", J. Math. Anal. Appl., 320, 902-915, 2006.
• Messaoudi S.A., Talahmeh A.A., Al-Shail J.H., "Nonlinear damped wave equation: Existence and blow-up", Comp. Math. Appl., 74, 3024-3041, 2017.
• Pişkin E., "Sobolev Spaces", Seçkin Publishing, 2017. (in Turkish).
• Ruzicka M., "Electrorheological Fluids: Modeling and Mathematical Theory", Lecture Notes in Mathematics, Springer, 2000.
• Song H., "Blow up arbitrarily positive inital energy solutions for a viscoelastic wave equation", Nonlinear Anal.: Real Worl Appl., 26, 306-314, 2015.
• Park S.H., Lee M.J., Kang J.R., "Blow up results for viscoelastic wave equations with weak damping", Appl. Math. Lett., 80, 20-26, 2018.