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Long-time behavior of second order evolution equations with nonlinear damping. (English) Zbl 1151.37059

Let $𝒜$ and $M$ be linear positive selfadjoint operators densely defend in a Hilbert space $H$. The authors consider Cauchy problem for the second order abstract equation

$\begin{array}{c}M{u}_{tt}\left(t\right)+𝒜u\left(t\right)+k·D\left({u}_{t}\left(t\right)\right)=F\left(u\left(t\right),{u}_{t}\left(t\right)\right),\\ {u|}_{t=0}={u}_{0}\in 𝒟\left({𝒜}^{1/2}\right),\phantom{\rule{4pt}{0ex}}{u}_{t}{|}_{t=0}={u}_{1}\in V=𝒟\left({M}^{1/2}\right)\end{array}\phantom{\rule{2.em}{0ex}}\left(1\right)$

under some additional hypotheses on the operators $D$ and $F$ guaranteeing the existence and uniqueness of a flow ${S}_{t}\left({u}_{0},{u}_{1}\right)\equiv \left(u\left(t\right),{u}_{t}\left(t\right)\right)$, $k$ is a positive parameter. Main attention is paid to flows generated by the problems with nonlinear dissipation $D$ and noncompact nonlinear part $F$, when the energy function is not necessarily decreasing. The interest to such problems is motivated by application to some models from continuum mechanics.

Chapter 1 introduces evolutions described by problems of type (1) and contains preliminary background material connected with quantitative problems of solutions to (1). Chapter 2 contains abstract results concerning the existence and properties of attractors in the context of general dynamical systems which are applied further in Chapter 3 to second-order evolutions given by (1). Chapters 3 and 4 provide general results on the following subjects: existence of global attractors; estimates of fractal dimension for attractors; regularity of elements on attractors; uniforms convergence rates (exponential, algebraic) of a single trajectory to an equilibrium and bounded sets to attractors; existence of exponential attractors (inertial sets) and existence of determining functionals.

The three last Chapters 5, 6 and 7 are devoted respectively to applications of developed the general theory: semilinear wave equation with nonlinear damping and local nonlinearity; von Karman system from elasticity theory with nonlinear dissipation (von Karman evolutions with rotational forces, characterized by finite speed of propagations and von Karman evolutions without rotational inertia, characterized by the infinite speed of propagation); several other examples from continuum mechanics, which demonstrate other types of nonlinearities and damping and include Berger, Mindlin-Timoshenko and Kirchhoff models of plates, together with systems with strong damping.

MSC:
 37L30 Attractors and their dimensions, Lyapunov exponents 34G20 Nonlinear ODE in abstract spaces 47H20 Semigroups of nonlinear operators 35B41 Attractors (PDE) 35L05 Wave equation (hyperbolic PDE) 37L05 General theory, nonlinear semigroups, evolution equations