Problems with variable exponents have attracted a great deal of attention lately and various existence, nonexistence and stability results have been established. The importance of such problems has manifested due to the recent advancement of science and technology and to the wide application in areas such as electrorheological fluids (smart fluids) which have the property that the viscosity changes drastically when exposed to heat or electrical fields. To tackle and understand these models, new sophisticated mathematical functional spaces have been introduced, such as the Lebesgue and Sobolev spaces with variable exponents. In this work, we are concerned with a system of wave equations with variable-exponent nonlinearities. This system can be regarded as a model for interaction between two fields describing the motion of two “smart” materials. We, first, establish the existence of global solutions then show that solutions of enough regularities stabilize to the rest state either exponentially or polynomially depending on the range of the variable exponents. We also present some numerical tests to illustrate our theoretical findings.
K.Agre and M.A.Rammaha, Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations.19(11) (2006), 1235–1270.
2.
S.Antontsev, Wave equation with -Laplacian and damping term: Existence and blow-up, Differ. Equ. Appl.3(4) (2011), 503–525.
3.
S.Antontsev, Wave equation with -Laplacian and damping term: Blow-up of solutions, C. R. Mec.339(12) (2011), 751–755. doi:10.1016/j.crme.2011.09.001.
4.
S.Antontsev and J.Ferreira, Existence, uniqueness and blowup for hyperbolic equations with nonstandard growth conditions, Nonlinear Anal. Theory Methods Appl.93 (2013), 62–77. doi:10.1016/j.na.2013.07.019.
5.
S.Antontsev and S.Shmarev, Evolution PDEs with nonstandard growth conditions: Existence, uniqueness, localization, blowup, in: Atlantis Studies in Differential Equations, Vol. 4, Atlantis Press, 2015.
6.
A.Benaissa and S.A.Messaoudi, Exponential decay of solutions of a nonlinearly damped wave equation, Nonlinear Differ. Equ. Appl.12(4) (2006), 391–399. doi:10.1007/s00030-005-0008-5.
7.
Y.Chen, S.Levine and M.Rao, Variable exponent, linear growth functionals in image restoration, SIAM journal on Applied Mathematics.66(4) (2006), 1383–1406. doi:10.1137/050624522.
8.
L.Diening, P.Harjulehto, P.Hasto and M.Ruzicka, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, 2011.
9.
D.Edmunds and J.Rakosnik, Sobolev embeddings with variable exponent, Studia Mathematica.143(3) (2000), 267–293. doi:10.4064/sm-143-3-267-293.
10.
D.Edmunds and J.Rakosnik, Sobolev embeddings with variable exponent. II, Mathematische Nachrichten.246(1) (2002), 53–67. doi:10.1002/1522-2616(200212)246:1<53::AID-MANA53>3.0.CO;2-T.
11.
X.Fan and D.Zhao, On the spaces and , Journal of Mathematical Analysis and Applications.263(2) (2001), 424–446. doi:10.1006/jmaa.2000.7617.
12.
S.Ghegal, I.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.
13.
B.Guo and W.Gao, Blow-up of solutions to quasilinear hyperbolic equations with -Laplacian and positive initial energy, C. R. Mec.342(9) (2014), 513–519. doi:10.1016/j.crme.2014.06.001.
14.
X.Han and M.Wang, Global existence and blow-up of solutions for a system of nonlinear viscoelastic wave equations with damping and source, Nonlinear Anal.71 (2009), 5427–5450. doi:10.1016/j.na.2009.04.031.
15.
A.M.Kbiri, S.A.Messaoudi and H.B.Khenous, A blow-up result for nonlinear generalized heat equation, Computers and Mathematics with Applications.68(12) (2014), 1723–1732. doi:10.1016/j.camwa.2014.10.018.
16.
V.Komornik, Decay estimates for the wave equation with internal damping, Int. Ser. Numer. Math.118 (1994), 253–266.
17.
M.T.Lacroix-Sonrier, Distrubutions Espace de Sobolev Application, Ellipses Edition Marketing S. A, 1998.
18.
I.Lasiecka, Stabilization of wave and plate-like equation with nonlinear dissipation on the boundary, J. Differ. Equ.79 (1989), 340–381. doi:10.1016/0022-0396(89)90107-1.
19.
I.Lasiecka and D.Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differ. Integral Equ.6(3) (1993), 507–533.
20.
X.Li, B.Guo and M.Liao, Asymptotic stability of solutions to quasilinear hyperbolic equations with variable sources, Computers and Mathematics with Applications.79(4) (2020), 1012–1022. doi:10.1016/j.camwa.2019.08.016.
21.
J.Lions, Quelques methodes de resolution des problemes aux limites non lineaires, 2nd edn, Dunod, Paris, 2002.
22.
P.Martinez, A new method to obtain decay rate estimates for dissipative systems with localized damping, Rev. Mat. Complut.12(1) (1999), 251–283.
23.
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 (2018), 1863–1875. doi:10.1016/j.camwa.2018.07.035.
24.
S.A.Messaoudi and A.Talahmeh, On wave equation: Review and recent results, Arabian J. of Math.7 (2018), 113–145. doi:10.1007/s40065-017-0190-4.
25.
S.A.Messaoudi and A.A.Talahmeh, A blow-up result for a nonlinear wave equation with variable-exponent nonlinearities, Appl. Anal.96(9) (2017), 1509–1515. doi:10.1080/00036811.2016.1276170.
26.
S.A.Messaoudi and A.A.Talahmeh, A blow-up result for a quasilinear wave equation with variable- exponent nonlinearities, Math Meth Appl Sci.40 (2017), 6976–6986. doi:10.1002/mma.4505.
27.
S.A.Messaoudi, A.A.Talahmeh and H.J.Al-Smail, Nonlinear damped wave equation: Existence and blow-up, Computers and Mathematics with Applications.74 (2017), 3024–3041. doi:10.1016/j.camwa.2017.07.048.
28.
M.I.Mustafa and S.A.Messaoudi, General energy decay rates for a weakly damped wave equation, Commun. Math. Anal.9(2) (2010), 67–76.
29.
M.Nakao, Decay of solutions of some nonlinear evolution equations, J. Math. Anal. Appl.60 (1977), 542–549. doi:10.1016/0022-247X(77)90040-3.
30.
M.Nakao, A difference inequality and its applications to nonlinear evolution equations, J. Math. Soc. Jpn.30 (1978), 747–762.
31.
M.Nakao, Remarks on the existence and uniqueness of global decaying solutions of the nonlinear dissipativewave equations, Math Z.206 (1991), 265–275. doi:10.1007/BF02571342.
32.
M.Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann.305(3) (1996), 403–417. doi:10.1007/BF01444231.
33.
N.M.Newmark, A method of computation for structural dynamics, Journal of the engineering mechanics division.85(3) (1959), 67–94. doi:10.1061/JMCEA3.0000098.
34.
B.Said-Houari, S.Messaoudi and A.Guesmia, General decay of solutions of a nonlinear system of viscoelastic wave equations, Nonlinear Differ. Equ. Appl.18 (2011), 659–684. doi:10.1007/s00030-011-0112-7.
35.
E.Zuazua, Exponential decay for the semi-linear wave equation with locally distributed damping, Comm. P.D.E.15 (1990), 205–235. doi:10.1080/03605309908820684.
