A Damped Nonlinear Hyperbolic Equation with Nonlinear Strain Term
DOI:
https://doi.org/10.5890/JAND.2022.03.010Abstract
In this work, we investigate an initial boundary value problem related to the nonlinear hyperbolic equation $u_{tt} + u_{xxxx} + \alpha u_{xxxxt} = f(u_x)_x$, for $f(s) = |s|^\rho + |s|^\sigma$, $1 < \rho, \sigma, \alpha > 0$. Under suitable conditions, we prove the existence of global solutions and the exponential decay of energy. The nonlinearity $f(s)$ introduces some obstacles in the process of obtaining a priori estimates and we overcome this difficulty by employing an argument due to Tartar (1978). The exponential decay is obtained via an integral inequality introduced by Komornik (1994).References
[1] Russell, D.L. and White, L.W. (2002), A Nonlinear Elastic Beam System with Inelastic Contact Constraints, Appl Math Optim., 46, 291-312.
[2] Yang, Z. (2003), Global existence, asymptotic behavior and blowup of solutions for a class of nonlinear wave equations with dissipative term, J. Differential Equations, 187, 520-540.
[3] Lian, W., Xu, R.Z., Rv{a}dulescu, V.D., Yang, Y.B., and Zhao, N. (2019), Global well-posedness for a class of fourth order nonlinear strongly damped wave equations, Adv. Calc. Var., http://dx.doi.org/10.1515/acv-2019-003.
[4] Tartar, L. (1978), Topics in Nonlinear Analysis, Publications Math{ematiques d'Orsay}, Uni.Paris Sud. Dep. Math., Orsay, France.
[5] Boukerrioua, K. and Guezane-Lakoud A. (2008), Some nonlinear integral inequalities arising in differential equations , Electron. J. Diff. Eqns., 80, 1-6.
[6] Ragusa, M.A. and Tachikawa, A. (2005), Partial regularity of the minimizers for quadratic functionals with VMO coefficients, J. Lond. Math. Soc.-Second Series, 72(3), 609-620.
[7] Komornik, V. (1994), Exact Controllability and Stabilization, Research in Applied Mathematics, Masson, Paris, France.
[8] Pang, T. and Shen, J. (2020), Global existence and asymptotic behaviour of solution for a damped nonlinear hyperbolic equation, Nonlinear Anal., 198, Article 111885.
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