On a biharmonic coupled system with non-standard nonlinearity: Existence,blow up and numerics

Abstract

In this paper, we consider a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities, supplemented with initial and mixed boundary conditions. We establish an existence and uniqueness result of a weak solution, under suitable assumptions on the variable exponents. Then, we show that solutions with negative-initial energy blow up in finite time. To illustrate our theoritical findings, we present two numerical examples.

Keywords

Biharmonic operator Existence Blow up Coupled system Variable exponent Weak solution

Get full access to this article

View all access options for this article.

References

Aassila and

Guesmia, Energy decay for a damped nonlinear hyperbolic equation, Appl. Math Lett. 12 (1999), 49–52. doi:10.1016/S0893-9659(98)00171-2.

Agre and

Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations. 19 (2006), 1235–1270.

Alves,

Cavalcanti,

Cavalcanty,

Rammaha and

Toundykov, On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms, Discrete and Continuous Dynamical Systems Series S 2(3) (2009), 583–608. doi:10.3934/dcdss.2009.2.583.

L.E.

Andersson,

Elfving and

G.H.

Golub, Solution of biharmonic equations with application to radar imaging, J. Comp. and Appl. Math. 94(2) (1998), 153–180. doi:10.1016/S0377-0427(98)00079-X.

Antontsev,

Ferreira and

Piskin, Existence and blow up of solutions for a strongly damped Petrovsky equation with variable-exponent nonlinearities, Electronic Journal of Differential Equations 2021 (2021), 1–18. doi:10.58997/ejde.2021.84.

Antontsev and

Shmarev, Blow-up of solutions to parabolic equations with nonstandard growth conditions, J. Comput. Appl. Math. 234(9) (2010), 2633–2645. doi:10.1016/j.cam.2010.01.026.

Antontsev and

Shmarev, Evolution PDEs with Nonstandard Growth Conditions, Atlantis Studies in Differential Equations, Vol. 4, Atlantis Press, Paris, 2015.

H.T.

Banks,

Ito and

Wang, Well-posedness for damped second order systems with unbounded input operators, Differential and Integral Equations. 8 (1995), 587–606.

M.I.G.

Bloor and

M.J.

Wilson, An approximate analytic solution method for the biharmonic problem, Proccedings of the Royal Society A: Mathiematical, Physical and Engineering Science. 462 (2006), 1107–1121.

10.

Bouhoufani and

Hamchi, Coupled system of nonlinear hyperbolic equations with variable-exponents: Global existence and Stability, Mediterranean Journal of Mathematics 17(166) (2020). doi:10.1007/s00009-020-01589-1.

11.

Bouhoufani,

S.A.

Messaoudi and

Zahri, Existence and decay of solutions for a coupled system of nonlinear wave equations with variable exponents, Elect. J. Differential Eqns. 73 (2023), 1–20.

12.

D.V.

Cruz-Urib and

Fiorenza, Variable Lebesgue Space: Foundations and Harmonic Analysis, Springer, Heidelberg New York Dordrecht London, 2013.

13.

Georgiev and

Todorova, Existence of solutions of the wave equation with nonlinear damping and source terms, J. Diff. Eqns. 109 (1994), 295–308. doi:10.1006/jdeq.1994.1051.

14.

Ghegal,

Hamchi and

S.A.

Messaoudi, Global existence and stability of a nonlinear wave equation with variable-exponent nonlinearities, Applicable Analysis. 99(8) (2020), 1333–1343. doi:10.1080/00036811.2018.1530760.

15.

Guesmia, Existence globale et stabilisation interne non linéaire d’un systeme de Petrovisky, Bell. Belg. Math. Soc. 5 (1998), 583–594.

16.

Guesmia, Energy decay for a damped nonlinear coupled system, J. Math. Anal. Appl. 239 (1999), 38–48. doi:10.1006/jmaa.1999.6534.

17.

Guo and

Gao, Blow up of solutions to quasilinear hyperbolic equations with

p (x, t)

-Laplacian and positive initial energy, C. R. Mecanique. 342 (2014), 513–519. doi:10.1016/j.crme.2014.06.001.

18.

J.H.

Hassan, General decay results for a viscoelastic wave equation with a variable exponent nonlinearity, in: Asymptotic Analysis., Vol. 125, IOS Press, 2021, pp. 365–388. doi:10.3233/ASY-201661.

19.

Komornik, Exact Controllability and Stabilization. the Multiplier Method, Masson, Paris, 1994.

20.

Komornik, Well-posedness and decay estimates for a Petrovesky system by a semigroup approach, Acta. Sci. Math. (Szeged) 60 (1995), 451–466.

21.

Lars,

Hasto and

Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Lect Notes Math, Springer, 2011.

22.

A.C.

Lazer and

P.J.

Mckenna, Large-amplitude periodique oscillations in suspension bridges: Some new connections with nonlinear analysis, Siam Review. 32(4) (1990), 537–578. doi:10.1137/1032120.

23.

Li,

Guo and

Liao, Asymptotic stabilty of solutions to quasilinear hyperbolic equations with variable sources, Computers and Mathematics with Applications. 79(4) (2000), 1012–1022. doi:10.1016/J.camwa.2019.08.016.

24.

Liao and

Tan, On behavior of solutions to a Petrovsky equation with damping and variable-exponent source, Math. App. (2021), 1–21.

25.

J.L.

Lions, Quelques Méthodes de Résolution des Problemes aux Limites Nonlinéaires, 2nd edn, Dunod, Paris, 2002.

26.

Messaoudi,

Bouhoufani,

Hamchi and

Alahyane, Existence and blow up in a system of wave equations with nonstandard nonlinearities, Electronic Journal of Differential Equations. 91 (2021), 1–33.

27.

Messaoudi and

Talahmeh, Blow up in solutions of a quasilinear wave equation with variable-exponent nonlinearities, Mathematical Methods in the Applied Sciences 40(18) (2017), 6976–6986. doi:10.1002/mma.4505.

28.

Messaoudi and

Talahmeh, Blow up of negative initial-energy solutions of a system of nonlinear wave equations with variable-exponent nonlinearities, Discrete and Continuous Dynamical Systems Series S, to appear.

29.

Messaoudi,

Talahmeh and

Al-Gharabli, On the existence and stability of a nonlinear wave system with variable exponents, in: Asymptotic Analysis., Vol. 1, IOS Press, 2021, pp. 1–28. doi:10.3233/ASY-211704.

30.

Messaoudi,

Talahmeh and

Al-Smail, Nonlinear damped wave equation: Existence and blow-up, Computers and Mathematics with applications. 74(12) (2017), 3024–3041. doi:10.1016/j.camwa.2017.07.048.

31.

S.A.

Messaoudi, Global existence and nonexistence in a system of Petrovsky, Journal of Mathematical Analysis and Applications. 265 (2002), 296–308. doi:10.1006/jmaa.2001.7697.

32.

S.A.

Messaoudi,

J.H.

Al-Smail and

A.A.

Talahmeh, Decay for solutions of a nonlinear damped wave equation with variable-exponent nonlinearities, Computers and Mathematics with applications. 76(8) (2018), 1863–1875. doi:10.1016/j.camwa.2018.07.035.

33.

S.-H.

Park and

J.-R.

Kang, Blow-up of solutions for a viscoelastic wave equation with variable exponents, Math. Meth. Appl. Sci. 42 (2019), 2083–2097. doi:10.1002/mma.5501.

34.

J.A.

Pelesko and

D.H.

Bernstein, Modeling Mems and Nems, CRC Press, 2002.

35.

Ružička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, Vol. 1748, Springer, Berling, Heidelberg, 2000.

36.

T.V.

Smitha,

K.V.

Nagaraja and

Sarada, Matlab 2D higher-order triangle mesh generator with finite element applications using subparametric transformations, Advances in Engineering Software 115 (2018), 327–356. doi:10.1016/j.advengsoft.2017.10.012.

37.

Timoshenko and

W.K.

Sergius, Theory of Plates and Shells, 2nd edn, McGraw-Hill, New York, 1995, 138317481406.

38.

Xiaolei,

Bin and

Menglan, Asymptotic stability of solutions to quasilinear hyperbolic equations with variable sources, Computers and Mathematics with Applications (2019), 1–11.