×

The dynamics of a nonlinear wave equation. (English) Zbl 1015.35072

Author’s abstract: We consider a wave equation in a bounded domain with linear dissipation and with a nonlinear source term. We give characterizations of all the solutions with respect to their qualitative properties: globality, boundedness, nonglobality, blow-up and convergence to equilibria.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B35 Stability in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Payne, L. E.; Sattinger, D. H., Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22, 273-303 (1975) · Zbl 0317.35059
[2] Ikehata, R., Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27, 1165-1175 (1996) · Zbl 0866.35071
[3] Cazenave, T., Uniform estimates for solutions of nonlinear Klein-Gordon equations, J. Funct. Anal., 60, 36-55 (1985) · Zbl 0568.35068
[4] Cazenave, T.; Haraux, A., An Introduction to Semilinear Evolution Equations (1998), Clarendon Press: Clarendon Press Oxford · Zbl 0926.35049
[5] Esquivel-Avila, J. A., Bounded solutions and blow-up in nonlinear wave equations, (Proceedings of the International Conference on Differential Equations, EQUADIFF 99, Berlin, Germany, Vol. 1 (2000), World Scientific: World Scientific Singapore), 339-341 · Zbl 0976.35043
[6] Georgiev, V.; Todorova, G., Existence of a solution of the wave equation with nonlinear damping and source terms, J. Differential Equations, 109, 295-308 (1994) · Zbl 0803.35092
[7] Lions, J. L., Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires (1969), Dunod: Dunod Paris · Zbl 0189.40603
[8] Ikehata, R.; Suzuki, T., Stable and unstable sets for evolution equations of parabolic and hyperbolic type, Hiroshima Math. J., 26, 475-491 (1996) · Zbl 0873.35010
[9] Esquivel-Avila, J. A., A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations, Nonlinear Anal., 52, 111-127 (2003) · Zbl 1023.35076
[10] Webb, G. F., Compactness of bounded trajectories of dynamical systems in infinite dimensional spaces, Proc. Roy. Soc. Edinburgh Sect. A, 84, 19-34 (1979) · Zbl 0414.34042
[11] Ball, J., Stability theory for extensible beam, J. Differential Equations, 14, 399-418 (1973) · Zbl 0247.73054
[12] Ball, J., On the asymptotic behaviour of generalized processes, with applications to nonlinear evolution equations, J. Differential Equations, 27, 224-265 (1978) · Zbl 0376.35002
[13] Haraux, A.; Jendoubi, M. A., Convergence of bounded weak solutions of the wave equation with dissipation and analytic nonlinearity, Calc. Var. Partial Differential Equations, 9, 95-124 (1999) · Zbl 0939.35122
[14] Jendoubi, M. A., Convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity, J. Differential Equations, 144, 302-312 (1998) · Zbl 0912.35028
[15] Haraux, A.; Jendoubi, M. A., Convergence of solutions of second-order gradient-like systems with analytic nonlinearities, J. Differential Equations, 144, 313-320 (1998) · Zbl 0915.34060
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.