Abstract
Long time behavior of a semilinear wave equation with nonlinear boundary dissipation is considered. It is shown that weak solutions generated by the wave dynamics converge asymptotically to a finite dimensional attractor. It is known [CEL1] that the attractor consists of all full trajectories emanating from the set of stationary points. Under the additional assumption that the set of stationary points is finite it is proved that every solution converges to some stationary points at an exponential rate.
Highlights
Long time behavior of a semilinear wave equation with nonlinear boundary dissipation is considered
Hyperbolic flows with nonlinear dissipation are not C1 -a feature that is fundamental to all treatments that are based on linearization of the flow [T,E-M-N,B-V]
Our main aim is to answer in affirmative the four questions raised above. This will be accomplished by combining recent observability estimates obtained in the context of control theory with some new developments in dynamical systems that pertain to Kolmogorov entropy and related fractal dimensions
Summary
Abstract: Long time behavior of a semilinear wave equation with nonlinear boundary dissipation is considered. ”squeezing” property - a fundamental tool in proving finite dimensionality [E-M-N,T,Lad,E-F-N-T]- requires a substantial amount of regularity of solutions in order to propagate the smoothness through the nonlinear dissipative term. Theorem 2.7 Under the Assumption 1 with g (s) > 0, s ∈ R, and p < 1 when n = 3 the dynamical system (S(t), H) admits a compact global attractor A whose fractal dimension is finite.
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