Introduction: Long-Time Behavior of Evolution Inclusion
Solutions in Earth Data Analysis ............................. xiii
References ................................................... xxxi
Part I Longtime Behavior of Autonomous Differential-
Operator Systems Solutions for Earth Data Processing
1 Abstract Theory of Multivalued Semiflows ..................... 3
1.1 oi-Limit Sets and Global Attractors of Multivalued
Semiflows ............................................... 5
1.2 Comparison Between Trajectory and Global Attractors
for Evolution Systems .................................. 17
1.2.1 The Main Definitions ............................ 18
1.2.2 Main Results .................................... 21
1.2.3 Some Model Applications ......................... 30
References .................................................. 33
2 Auxiliary Properties of Evolution Inclusions Solutions
for Earth Data Processing ................................... 37
2.1 Preliminaries .......................................... 38
2.2 Pointwise Pseudomonotone Maps .......................... 43
2.3 Auxiliary Properties of Solutions for the First Order
Evolution .............................................. 53
2.3.1 The Setting of the Problem ...................... 53
2.3.2 Preliminaries ................................... 54
2.3.3 Supplementary Properties of Solutions ........... 56
2.4 Asymptotic Behavior of the First Order Evolution
Inclusions ............................................. 61
2.4.1 Existence of the Global Attractor ............... 62
2.4.2 Existence of the Trajectory Attractor ........... 63
2.4.3 Comments ........................................ 66
2.4.4 Conclusion ...................................... 67
2.5 Auxiliary Properties of Solutions for the Second
Order Evolution Inclusions ............................. 67
2.5.1 Preliminary Results ............................. 69
2.5.2 Auxiliary Properties of the Resolving Operator .. 71
2.6 Auxiliary Properties of Solutions for the Second
Order Evolution Inclusions ............................. 79
2.7 Asymptotic Behavior of the Second-Order Evolution
Inclusions ............................................. 86
2.7.1 Existence of the Global Attractor ............... 87
2.7.2 Existence of the Trajectory Attractor ........... 93
2.7.3 Auxiliary Properties of the Global and
Trajectory Attractors ........................... 96
2.8 Applications .......................................... 100
2.8.1 Climate Energy Balance Model ................... 100
2.8.2 Application for General Classes High-Order
Nonlinear PDEs ................................. 101
2.8.3 Application for Chemotaxis Processes ........... 102
2.8.4 Applications for Damped Viscoelastic Fields
with Short Memory .............................. 105
2.8.5 Applications for Nonsmooth Autonomous
Piezoelectric Fields ........................... 110
References ................................................. 116
3 Attractors for Lattice Dynamical Systems ................... 119
3.1 Existence of Solutions ................................ 120
3.2 A Priori Estimates .................................... 141
3.2.1 Existence of a Bounded Absorbing Set ........... 141
3.2.2 Estimate of the Tails .......................... 142
3.3 Existence of the Global Attractor ..................... 144
3.4 Approximation of the Attractor ........................ 151
3.5 Application for Discrete Climate Energy Balance
Model ................................................. 158
References ................................................. 159
Part II Longtime Behavior of Nonautonomous Differential-
Operator Systems Solutions for Earth Data Processing
4 On Global Attractors of Multivalued Semiprocesses and
Nonautonomous Evolution Inclusions ......................... 163
4.1 ω-Limit Sets and Global Attractors of Multivalued
Semiprocesses ......................................... 164
4.2 Global Attractors for Nonautonomous Differential
Inclusions ............................................ 177
4.2.1 Abstract Setting: Construction of the
Multivalued Semiprocess ........................ 177
4.2.2 Global Attractors of Nonautonomous Reaction-
Diffusion Inclusions ........................... 182
4.3 Applications for Chemical Kinetics Processes and
Fields ................................................ 197
References ................................................. 197
5 On the Kneser's Property for the Complex Ginzburg-Landau
Equation and the Lotka-Volterra System with Diffusion ...... 199
5.1 Setting of the Problem ................................ 202
5.2 The Kneser's Property for Reaction-Diffusion
Systems ............................................... 204
5.2.1 Application to the Complex Ginzburg-Landau
Equation ....................................... 220
5.2.2 Application to the Lotka-Volterra System
with Diffusion ................................. 221
5.3 Connectedness of Attractors for Reaction-Diffusion
Systems ............................................... 222
5.3.1 Application to the Complex Ginzburg-Landau
Equation ....................................... 227
5.3.2 Application to the Lotka-Volterra System with
Diffusion ...................................... 228
References ................................................. 229
6 Pullback Attractors for a Class of Extremal Solutions of
the 3D Navier-Stokes System ................................ 231
6.1 Pullback Attractors for Multivalued Processes ......... 232
6.2 Setting of the Problem and Main Results ............... 235
6.3 Relationship with the Attractor of the 3D Navier-
Stokes System ......................................... 250
6.4 Applications for Hydrodynamic Problems ................ 253
References ................................................. 256
7 Properties of Resolving Operator for Nonautonomous
Evolution Inclusions ....................................... 259
7.1 New Theorems for Existence Solutions for Skrypnik's
Type Operators ............................................. 261
7.1.1 Problem Definition ............................. 262
7.1.2 The Class  (χ*) ................................ 264
7.1.3 Classes of Multivalued Maps .................... 264
7.1.4 The Main Results ............................... 267
7.2 Noncoercive Evolution Inclusions for Sk Type
Operators ............................................. 276
7.2.1 Preliminaries: On Some Classes of Multivalued
Maps ........................................... 279
7.2.2 Setting of the Problem ......................... 281
7.2.3 Main Results ................................... 285
7.2.4 Applications ................................... 298
7.3 Functional-Topological Properties of the Resolving
Operator for the Evolution Inclusion .................. 300
7.3.1 The Setting of the Problem ..................... 301
7.3.2 Main Results ................................... 301
7.4 Auxiliary Properties of Solutions for the
Nonautonomous First-Order Evolution Inclusions with
Uniformly Coercive Mappings, Long-Time Behavior, and
Pullback Introduction: Long-Time Behavior of
Evolution Inclusion Solutions in Earth Data Analysis
Attractors ............................................ 313
7.5 Applications .......................................... 317
References ................................................. 317
A Functional Spaces: The Embedding and Approximation
Theorems ................................................... 321
References ................................................. 328
Index ......................................................... 329
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