 | Chueshov I. Long-time behavior of second order evolution equations with nonlinear damping / I.Chueshov, I.Lasiecka. - Providence: American Mathematical Society, 2008. - viii, 183 p. - (Memoirs of the American Mathematical Society; Vol.195, N 912). - Bibliogr.: p.179-182. - Ind.: p.183. - ISBN 978-0-8218-4187-7; ISSN 0065-9266
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Preface ...................................................... viii
Chapter 1. Introduction ......................................... 1
1.1. Description of the problem studied ...................... 1
1.2. The model and basic assumption .......................... 4
1.3. Well-posedness .......................................... 8
Chapter 2. Abstract results on global attractors ............... 17
2.1. Criteria for asymptotic smoothness of dynamical
systems ................................................ 18
2.2. Criteria for finite dimensionality of attractors ....... 22
2.3. Exponentially attracting positively invariant sets ..... 28
2.4. Gradient systems ....................................... 32
Chapter 3. Existence of compact global attractors for
evolutions of the second order in time .............. 38
3.1. Ultimate dissipativity ................................. 39
3.2. Asymptotic smoothness: the main assumption ............. 53
3.3. Global attractors in subcritical case .................. 56
3.4. Global attractors in critical case ..................... 63
Chapter 4. Properties of global attractors for evolutions of
the second order in time ............................ 90
4.1. Finite dimensionality of attractors .................... 90
4.2. Regularity of elements from attractors ................ 101
4.3. Rate of stabilization to equilibria ................... 113
4.4. Determining functionals ............................... 120
4.5. Exponential fractal attractors (inertial sets) ........ 122
Chapter 5. Semilinear wave equation with a nonlinear
dissipation ........................................ 125
5.1. The model ............................................. 125
5.2. Main results .......................................... 127
5.3. Proofs ................................................ 132
Chapter 6. Von Karman evolutions with a nonlinear
dissipation ........................................ 140
6.1. The model ............................................. 140
6.2. Properties of von Karman bracket ...................... 141
6.3. Abstract setting of the model ......................... 142
6.4. Model with rotational forces: α > 0 ................... 144
6.5. Non-rotational case α = 0 ............................. 152
Chapter 7. Other models from continuum mechanics .............. 158
7.1. Berger's plate model .................................. 158
7.2. Mindlin-Timoshenko plates and beams ................... 164
7.3. Kirchhoff limit in Mindlin-Timoshenko plates and
beams ................................................. 167
7.4. Systems with strong damping ........................... 173
Bibliography .................................................. 179
Index ......................................................... 183
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