# Blow up of Solution for a Nonlinear Viscoelastic Problem with Internal Damping and Logarithmic Source Term

### Abstract

This paper is concerned with blow up of weak solutions of the following nonlinear viscoelastic problem with internal damping and logarithmic source term |ut|ρutt + M(∥u∥2)(-∆u) - ∆utt + Z0t g(t - s)∆u(s)ds + ut = u|u|p R-2 ln |u|k R with Dirichlet boundary initial conditions in a bounded domain Ω ⊂ Rn. In the physical point of view, this is a type of problems that usually arises in viscoelasticity. It has been considered with power source term first by Dafermos [3], in 1970, where the general decay was discussed. We establish conditions of p, ρ and the relaxation function g, for that the solutions blow up in finite time for positive and nonpositive initial energy. We extend the result in [15] where is considered M = 1 and external force type |u|p-2u in it. Further we state and sketch the proof of a result of local existence of weak solution that is used in the proof of the theorem on blow up. The idea underlying the proof of local existence of solution is based on Faedo-Galerkin method combined with the Banach fixed point method.### References

[1] Cavalcanti M.M., Domingos Cavalcanti V.N., Ferreira J. Existence and uniform decay for a non-linear viscoelastic equation with strong damping, Mathematical Methods In Applied Sciences. 24(14), (2001): 1043–1053.

[2] Cordeiro S.M.S., Pereira D.C., Ferreira J., Raposo C.A. Global solutions and exponential decay to a Klein-Gordon equation of Kirchhoff-Carrier type with strong damping and nonlinear logarithmic source term, Partial Differential Equations in Applied Mathematics, Vol.3, (2021):6 p.

[3] Dafermos C. M. Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37, (1970): 297–308.

[4] Liu, W., Li, G., Hong, L. General Decay and Blow-Up of Solutions for a System of Viscoelastic Equations of Kirchhoff Type with Strong Damping, Journal of Function Spaces, (2014): 21 p.

[5] Lions J.-L. On some questions in boundary value problems of mathematical physics, IM-UFRJ, (1978).

[6] Liu W. Global existence, asymptotic behavior and blow-up of solutions for a viscoelastic equation with strong damping and nonlinear source, Topological Methods in Nonlinear Analysis, Vol. 36, no. 1, (2010): 153–178.

[7] Love A.H.A treatise on the mathematical theory of elasticity, Dover New York, (1944).

[8] Messaoudi S. A. Blow up and global existence in a nonlinear viscoelastic wave equation, Mathematische Nachrichten, Vol.260 (1), (2003): 58–66.

[9] Mezoua N., Mahmoud Boulaaras S.M., Allahem A. Global Existence of Solutions for the Viscoelastic Kirchhoff Equation with Logarithmic Source Terms, Hindawi Complexity, (2020), 25 p.

[10] Nishihara K. On a global solution of some quasilinear hyperbolic equation, Tokyo Journal of Math., Vol. 7, (1984):437–459.

[11] Pereira D.C., Izaguirre R.M. Sobre uma equa¸c˜aes de vibra¸c˜oes n˜ao lineares, Proceedings of 23 seminario brasileiro de Analise, Campinas, Sao Paulo, (1986): 155-158.

[12] Pohozaev S.I. On a class of quasillnear hyperbolic equations. Mat USSR Sbornik, Vol.25, no. 1, (1975): 145–148.

[13] Salim A.M., Tatar N. Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Mathematical Methods In The Applied Sciences, Vol. 30, (2007):665–680.

[14] Salim A.M., Tatar N. Global Existence and Asymptotic Behavior for a Nonlinear Viscoelastic Problem, Mathematical Sciences Research Journal, Vol. 7, no. 4, (2003): 136–149.

[15] Wu ST. Blow-Up of Solution for A Viscoelastic Wave Equation with Delay, Acta Math Sci., 39, (2019): 329–338.

[2] Cordeiro S.M.S., Pereira D.C., Ferreira J., Raposo C.A. Global solutions and exponential decay to a Klein-Gordon equation of Kirchhoff-Carrier type with strong damping and nonlinear logarithmic source term, Partial Differential Equations in Applied Mathematics, Vol.3, (2021):6 p.

[3] Dafermos C. M. Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37, (1970): 297–308.

[4] Liu, W., Li, G., Hong, L. General Decay and Blow-Up of Solutions for a System of Viscoelastic Equations of Kirchhoff Type with Strong Damping, Journal of Function Spaces, (2014): 21 p.

[5] Lions J.-L. On some questions in boundary value problems of mathematical physics, IM-UFRJ, (1978).

[6] Liu W. Global existence, asymptotic behavior and blow-up of solutions for a viscoelastic equation with strong damping and nonlinear source, Topological Methods in Nonlinear Analysis, Vol. 36, no. 1, (2010): 153–178.

[7] Love A.H.A treatise on the mathematical theory of elasticity, Dover New York, (1944).

[8] Messaoudi S. A. Blow up and global existence in a nonlinear viscoelastic wave equation, Mathematische Nachrichten, Vol.260 (1), (2003): 58–66.

[9] Mezoua N., Mahmoud Boulaaras S.M., Allahem A. Global Existence of Solutions for the Viscoelastic Kirchhoff Equation with Logarithmic Source Terms, Hindawi Complexity, (2020), 25 p.

[10] Nishihara K. On a global solution of some quasilinear hyperbolic equation, Tokyo Journal of Math., Vol. 7, (1984):437–459.

[11] Pereira D.C., Izaguirre R.M. Sobre uma equa¸c˜aes de vibra¸c˜oes n˜ao lineares, Proceedings of 23 seminario brasileiro de Analise, Campinas, Sao Paulo, (1986): 155-158.

[12] Pohozaev S.I. On a class of quasillnear hyperbolic equations. Mat USSR Sbornik, Vol.25, no. 1, (1975): 145–148.

[13] Salim A.M., Tatar N. Global existence and uniform stability of solutions for a quasilinear viscoelastic problem, Mathematical Methods In The Applied Sciences, Vol. 30, (2007):665–680.

[14] Salim A.M., Tatar N. Global Existence and Asymptotic Behavior for a Nonlinear Viscoelastic Problem, Mathematical Sciences Research Journal, Vol. 7, no. 4, (2003): 136–149.

[15] Wu ST. Blow-Up of Solution for A Viscoelastic Wave Equation with Delay, Acta Math Sci., 39, (2019): 329–338.

How to Cite

FERREIRA, J. et al.
Blow up of Solution for a Nonlinear Viscoelastic Problem with Internal Damping and Logarithmic Source Term.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 116, n. 4, dec. 2022. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/1194>. Date accessed: 28 jan. 2023. doi: https://doi.org/10.26577/JMMCS.2022.v116.i4.02.
Section

Mathematics

Keywords
Nonlinear Viscoelastic Equation, Logarithmic Source, Blow Up, Local existence.