Ye, Yaojun Exponential decay of energy for some nonlinear hyperbolic equations with strong dissipation. (English) Zbl 1211.35052 Adv. Difference Equ. 2010, Article ID 357404, 12 p. (2010). Summary: The initial boundary value problem for a class of hyperbolic equations with strong dissipative term \[ u_{tt}- \sum_{i=1}^n (\partial/\partial x_i) (|\partial u/\partial x_i|^{p-2} (\partial u/\partial x_i))- a\Delta u_t= b|u|^{r-2}u \]in a bounded domain is studied. The existence of global solutions for this problem is proved by constructing a stable set in \(W_0^{1,p}(\Omega)\) and showing the exponential decay of the energy of global solutions through the use of an important lemma of V. Komornik. Cited in 4 Documents MSC: 35B40 Asymptotic behavior of solutions to PDEs 35L72 Second-order quasilinear hyperbolic equations 35L20 Initial-boundary value problems for second-order hyperbolic equations Keywords:stable set; decay of energy PDFBibTeX XMLCite \textit{Y. Ye}, Adv. Difference Equ. 2010, Article ID 357404, 12 p. (2010; Zbl 1211.35052) Full Text: DOI EuDML References: [1] Andrews G:On the existence of solutions to the equation[InlineEquation not available: see fulltext.]. Journal of Differential Equations 1980,35(2):200-231. 10.1016/0022-0396(80)90040-6 · Zbl 0415.35018 [2] Andrews G, Ball JM: Asymptotic behaviour and changes of phase in one-dimensional nonlinear viscoelasticity.Journal of Differential Equations 1982,44(2):306-341. 10.1016/0022-0396(82)90019-5 · Zbl 0501.35011 [3] Ang DD, Dinh PN: Strong solutions of quasilinear wave equation with non-linear damping.SIAM Journal on Mathematical Analysis 1985, 19: 337-347. 10.1137/0519024 · Zbl 0662.35072 [4] Kawashima S, Shibata Y: Global existence and exponential stability of small solutions to nonlinear viscoelasticity.Communications in Mathematical Physics 1992,148(1):189-208. 10.1007/BF02102372 · Zbl 0779.35066 [5] Haraux A, Zuazua E: Decay estimates for some semilinear damped hyperbolic problems.Archive for Rational Mechanics and Analysis 1988,100(2):191-206. 10.1007/BF00282203 · Zbl 0654.35070 [6] Kopáčková M: Remarks on bounded solutions of a semilinear dissipative hyperbolic equation.Commentationes Mathematicae Universitatis Carolinae 1989,30(4):713-719. · Zbl 0707.35026 [7] Ball JM: Remarks on blow-up and nonexistence theorems for nonlinear evolution equations.The Quarterly Journal of Mathematics. Oxford 1977,28(112):473-486. 10.1093/qmath/28.4.473 · Zbl 0377.35037 [8] Yang Z: Existence and asymptotic behaviour of solutions for a class of quasi-linear evolution equations with non-linear damping and source terms.Mathematical Methods in the Applied Sciences 2002,25(10):795-814. 10.1002/mma.306 · Zbl 1011.35121 [9] Yang Z, Chen G: Global existence of solutions for quasi-linear wave equations with viscous damping.Journal of Mathematical Analysis and Applications 2003,285(2):604-618. 10.1016/S0022-247X(03)00448-7 · Zbl 1060.35087 [10] Zhijian Y: Initial boundary value problem for a class of non-linear strongly damped wave equations.Mathematical Methods in the Applied Sciences 2003,26(12):1047-1066. 10.1002/mma.412 · Zbl 1106.35318 [11] Yacheng L, Junsheng Z: Multidimensional viscoelasticity equations with nonlinear damping and source terms.Nonlinear Analysis: Theory, Methods & Applications 2004,56(6):851-865. 10.1016/j.na.2003.07.021 · Zbl 1057.74007 [12] Yang Z: Blowup of solutions for a class of non-linear evolution equations with non-linear damping and source terms.Mathematical Methods in the Applied Sciences 2002,25(10):825-833. 10.1002/mma.312 · Zbl 1009.35051 [13] Nakao M: A difference inequality and its application to nonlinear evolution equations.Journal of the Mathematical Society of Japan 1978,30(4):747-762. 10.2969/jmsj/03040747 · Zbl 0388.35007 [14] Ye Y: Existence of global solutions for some nonlinear hyperbolic equation with a nonlinear dissipative term.Journal of Zhengzhou University 1997,29(3):18-23. · Zbl 0899.35062 [15] Ye Y: On the decay of solutions for some nonlinear dissipative hyperbolic equations.Acta Mathematicae Applicatae Sinica. English Series 2004,20(1):93-100. 10.1007/s10255-004-0152-4 · Zbl 1070.35007 [16] Sattinger DH: On global solution of nonlinear hyperbolic equations.Archive for Rational Mechanics and Analysis 1968, 30: 148-172. 10.1007/BF00250942 · Zbl 0159.39102 [17] Komornik V: Exact Controllability and Stabilization, Research in Applied Mathematics. Masson, Paris, France; 1994:viii+156. · Zbl 0937.93003 [18] Ye Y: Existence and nonexistence of global solutions of the initial-boundary value problem for some degenerate hyperbolic equation.Acta Mathematica Scientia 2005,25(4):703-709. · Zbl 1090.35127 [19] Gao H, Ma TF:Global solutions for a nonlinear wave equation with the[InlineEquation not available: see fulltext.]-Laplacian operator.Electronic Journal of Qualitative Theory of Differential Equations 1999, (11):1-13. [20] Adams RA: Sobolev Spaces, Pure and Applied Mathematics. Volume 6. Academic Press, New York, NY, USA; 1975:xviii+268. [21] Ladyzhenskaya OA: The Boundary Value Problems of Mathematical Physics, Applied Mathematical Sciences. Volume 49. Springer, New York, NY, USA; 1985:xxx+322. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.