Attractors for second-order evolution equations with a nonlinear damping
Let \(\Omega\subset {\mathbb R}^n,n=2,3\), be a bounded connected set with a smooth boundary \(\Gamma\). The equation (1) \(\omega '-\Delta \omega +kg(\omega_t )+f(\omega )=0\) in \(Q=[0,\infty )\times \Omega\) with some boundary condition is considered. Here, \(g\in C^1 ({\mathbb R})\) is a monotone increasing function with some additional properties and \(f\in C^1 ({\mathbb R})\) is such that \(| f'(s)| \leq M| s| ^q, | s| \geq 1\) for some \(q\). It is shown that equation (1) generates a continuous semiflow \(S_t \) in \(H=H^1 (\Omega )\times L_2 (\Omega )\) and under some conditions, the dynamical system \((H,S_t )\) possesses a global compact attractor \({\mathcal A}\), under some hypotheses having a finite fractal dimension. There is a theorem dealing with the convergence to an equilibrium. Analogous results for a nonlinear system of dynamical elasticity described by von Karman evolution with a nonlinear dissipation are obtained. Main abstract results on global attractors are the following: 1. Let \((X,S_t )\) be a dissipative dynamical system in a Banach space \(X\). Then \((X,S_t )\) possesses a compact global attractor \({\mathcal A}\) iff \((X,S_t )\) is asymptotically smooth. 2. Criteria for asymptotic smoothness and criteria for finite dimensionality of attractors are obtained. 3. The long-time dynamics of abstract nonlinear second-order evolution equations with a nonlinear damping is studied. It is shown that any solution is stabilized to an equilibrium and the rate of convergence is estimated which in turn, depends on the behavior at the origin of the function describing the dissipation. If the damping is bounded from below by a linear function, this rate is exponential.
scientific article; zbMATH DE number 2145900
scientific article