Global existence, uniform decay, and exponential growth of solutions for a system of viscoelastic Petrovsky equations

In this paper, we study the initial-boundary value problem for a system of nonlinear viscoelastic Petrovsky equations. Introducing suitable perturbed energy functionals and using the potential well method we prove uniform decay of solution energy under some restrictions on the initial data and the relaxation functions. Moreover, we establish a growth result for certain solutions with positive initial energy.

Anahtar Kelimeler:

Global existence, uniform decay, exponential growth, viscoelastic Petrovsky equation

Global existence, uniform decay, and exponential growth of solutions for a system of viscoelastic Petrovsky equations

Keywords:

Global existence, uniform decay, exponential growth, viscoelastic Petrovsky equation,

PDF

___

Agre, K., Rammaha, M.A.: Systems of nonlinear wave equations with damping and source terms, Differential Integral Equations. 19(11), 1235–1270 (2006).
Alves, C.O., Cavalcanti, M.M., Domingos Cavalcanti, V.N., Rammaha, M. A., Toundykov, D.: On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms, Discrete Contin. Dyn. Syst. Ser. S 2(3), 583–608 (2009).
Amroun, N.E., Benaissa, A.: Global existence and energy decay of solutions to a Petrovsky equation with general nonlinear dissipation and source term. Georgian. Math. J. 13, 397–410 (2006).
Berrimi, S., Messaoudi, S.A.: Existence and decay of solutions of a viscoelastic equation with a nonlinear source. Nonlinear Anal. 64, 2314–2331 (2006).
Berrimi, S., Messaoudi, S.A.: Exponential decay of solutions to a viscoelastic equation with nonlinear localized damping. Electron. J. Differential Equations. 88, 1–10 (2004).
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Ferreira, J.: Existence and uniform decay for a nonlinear viscoelastic equation with strong damping. Math. Meth. Appl. Sci. 24, 1043–1053 (2001).
Cavalcanti, M.M., Oquendo, H.P.: Frictional versus viscoelastic damping in a semilinear wave equation. SIAM. J. Control. Optim. 42(4), 1310–1324 (2003).
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Soriano, J.A.: Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping. Electron. J. Differential Equations. 2002(44), 1–14 (2002). Cavalcanti, M.M., Domingos Cavalcanti, V.N.: Existence and asymptotic stability for evolution problems on manifolds with damping and source terms. J. Math. Anal. Appl. 291(1), 109–127 (2004).
Cavalcanti, M.M., Domingos Cavalcanti, V.N., Ma, T.F., Soriano, J.A.: Global existence and asymptotic stability for viscoelastic problems. Differential Integral Equations. 15 (6), 731–748 (2002).
Guesmia, A.: Existence globale et stabilisation interne non lin´ eaire d’un syst` eme de Petrovsky. Bell. Belg. Math. Soc. 5, 583–594 (1998).
Han, X.S., Wang, M.X.: Global existence and uniform decay for a nonlinear viscoelastic equation with damping. Nonlinear Anal. 70, 3090–3098 (2009).
Han, X.S., Wang, M.X.: General decay of energy for a viscoelastic equation with nonlinear damping. Math. Meth. Appl. Sci. 32(3), 346–358 (2009).
Liang, F., Gao, H.: Exponential energy decay and blow-up of solutions for a system of nonlinear viscoelastic wave equations with strong damping. Boundary Value Problems 2011. 22, (2011).
Li, M.R., Tsai, L.Y.: On a system of nonlinear wave equations. Taiwanese J. Math. 7, 557–573 (2003).
Li, G., Sun, Y., Liu, W.: Global existence and blow up of solutions for a strongly damped Petrovsky system with nonlinear damping. Appl. Anal. 91(3), 575–586 (2012).
Li, G., Sun, Y., Liu, W.: Global existence, uniform decay and blow up of solutions for a system of Petrovsky equations. Nonlinear Anal. 74, 1523–1538 (2011).
Liu, W.J.: General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source. Nonlinear Anal. 73(6), 1890–1904 (2010).
Liu, W.J.: General decay of solutions to a viscoelastic wave equation with nonlinear localized damping. Annal. Academ. Scien. Fenni. Math. 34, 291–306 (2009).
Liu, W.J.: Global existence, asymptotic behavior and blow-up of solutions for a viscoelastic equation with strong damping and nonlinear source. Topol. Methods Nonlinear Anal. 36(1), 153–178 (2010).
Liu, W.J.: Uniform decay of solutions for a quasilinear system of viscoelastic equations. Nonlinear Anal. 71, 2257–2267 (2010).
Ma, J., Mu, C., Zeng, R.: A blow up result for viscoelastic equations with arbitrary positive initial energy. Boundary Value Problems 2011. 6, (2011).
Messaoudi, S.A., Tatar, N-E.: Global existence and asymptotic behavior for a nonlinear viscoelastic problem. Math. Sci. Res. J. 7(4), 136–149 (2003).
Messaoudi, S.A., Tatar, N-E.: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem. Math. Meth. Appl. Sci. 30, 665–680 (2007).
Messaoudi, S.A., Tatar, N-E.: Uniform stabilization of solutions of a nonlinear system of viscoelastic equations. Appl. Anal. 87(3), 247–263 (2008).
Messaoudi, S.A., Said-Houari, B.: Global nonexistence of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. J. Math. Anal. Appl. 365, 277–287 (2010).
Messaoudi, S.A.: Global existence and nonexistence in a system of Petrovsky. J. Math. Anal. Appl. 265, 296–308 (2002).
Nakao, M.: Asymptotic stability of the bounded or almost periodic solution of the wave equation. J. Math. Anal. Appl. 56, 336–343 (1977).
Said-Houari, B.: Global existence and decay of solutions of a nonlinear system of wave equations. Appl. Anal. 91(3), 475–489 (2012).
Said-Houari, B.: Global nonexistence of positive initial-energy solutions of a system of nonlinear wave equations with damping and source terms. Differential Integral Equations. 23(1–2), 79–92 (2010).
Said-Houari, B.: Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms. Z. Angew. Math. Phys. 62, 115–133 (2011).
Said-Houari, B., Messaoudi, S.A., Guesmia, A.: General decay of solutions of a nonlinear system of viscoelastic wave equations. Nonlinear. Differ. Equ. Appl. 18, 659–684 (2011).
Vitillaro, E.: Global existence theorems for a class of evolution equations with dissipation. Arch. Ration. Mech. Anal. 149, 155–182 (1999).
Wu, S.T.: General decay of solutions for a viscoelastic equation with nonlinear damping and source terms. Acta. Math. Scientia. 31B, 1436–1448 (2011).
Wu, S.T.: General decay of energy for a viscoelastic equation with damping and source term. Taiwanese J. Math. 16(1), 113–128 (2012).
Wu, S.T.: Blow-up of solutions for a system of nonlinear wave equations with nonlinear damping. Electron. J. Differential Equations. 2009(105), 1–11 (2009).
Xu, R., Yang, Y., Liu, Y.: Global well-posedness for strongly damped viscoelastic wave equation. Appl. Anal. 92(1), 138–157 (2013).
Yu, S.Q.: On the strongly damped wave equation with nonlinear damping and source terms. Electron. J. Qual. Theory Differ. Eqns. 39, 1–18 (2009).