Attractors for damped nonlinear hyperbolic equations. (English) Zbl 0572.35071

Ghidaglia, J. M.; Temam, R.

J. Math. Pures Appl., IX. Sér. 66, 273-319 (1987).

This work is devoted to the study of the longtime behavior of damped hyperbolic equations. We show that this behavior is finite dimensional by proving that the attractors associated with these equations have finite fractal dimension. This last result is based on estimates of the Lyapunov exponents on the attractors.
In the first part we make some remarks and give an abstract setting that contains, in particular, the sine-Gordon equation and the nonlinear wave equation of quantum mechanics. In the second part we investigate the properties of the attractors including their dimension. And finally in the third part we give several examples and deal with a generalization to non-autonomous systems. In this last section we give a bound totaly explicit (in terms of the physical quantities) for the dimension of the maxmal attractor associated to the sine-Gordon equation with a forcing term which oscillates periodically in time.

MSC:

35L70	Second-order nonlinear hyperbolic equations
35B40	Asymptotic behavior of solutions to PDEs
35L20	Initial-boundary value problems for second-order hyperbolic equations
35Q99	Partial differential equations of mathematical physics and other areas of application

Keywords:

oscillates periodically; longtime behavior; damped hyperbolic equations; finite dimensional; attractors; finite fractal dimension; Lyapunov exponents; sine-Gordon equation; nonlinear wave equation; quantum mechanics