Preface ........................................................ xi
1. Preliminary
1.1. Sobolev Spaces and Their Basic Properties ............... 1
1.1.1. Distributions .................................... 2
1.1.2. Weak Derivatives and Sobolev Spaces .............. 4
1.1.3. Sobolev Inequalities, Embedding Theorems
and the Trace Theorem ............................ 9
1.1.4. Interpolation Inequalities ...................... 17
1.1.5. The Poincare Inequality ......................... 17
1.2. Some Inequalities in Analysis .......................... 18
1.2.1. The Classical Bellman-Gronwall Inequality ....... 18
1.2.2. The Generalized Bellman-Gronwall Inequalities ... 19
1.2.3. The Uniform Bellman-Gronwall Inequality ......... 20
1.2.4. The Nakao Inequalities .......................... 23
1.3. Some Differential Inequalities for Nonexistence of
Global Solutions ....................................... 25
1.4. Other Useful Inequalities .............................. 26
1.4.1. The Young Inequalities .......................... 27
1.4.2. The Holder Inequality ........................... 28
1.4.3. The Minkowski Inequalities ...................... 29
1.4.4. The Jensen Inequality ........................... 30
1.5. Co-Semigroups of Linear Operators ...................... 31
1.5.1. C0-Semigroups of Linear Operators ............. 31
1.6. Global Attractors ...................................... 37
1.6.1. Compact Semigroups (Semiflows) for Autonomous
Systems ......................................... 39
1.6.2. Weakly Compact Semigroups (Semiflows) for
Autonomous Systems .............................. 42
1.7. Bibliographic Comments ................................. 43
2. A One-dimensional Nonlinear Viscous and Heat-conductive
Real Gas
2.1. Fixed and Thermally Insulated Boundary Conditions ...... 46
2.1.1. Main Results .................................... 46
2.1.2. Uniform A Priori Estimates ...................... 49
2.2. Clamped and Constant Temperature Boundary Conditions ... 71
2.3. Exponential Stability in Hl and H2 ..................... 78
2.3.1. Main Results .................................... 78
2.3.2. Exponential Stability in Hl ..................... 80
2.3.3. Exponential Stability in H2 ..................... 89
2.4. Exponential Stability in H4 ............................ 97
2.4.1. Global Existence in H4 ......................... 100
2.4.2. A Nonlinear Co-Semigroup S(t) on H4 ............ Ill
2.4.3. Exponential Stability in H4 .................... 119
2.5. Attractors in Hl and H2 ............................... 123
2.5.1. An Absorbing Set in Hl ......................... 126
2.5.2. An Absorbing Set in H2 ......................... 132
2.6. Universal Attractor in H4 ............................. 135
2.7. Bibliographic Comments ................................ 138
3. A One-dimensional Polytropic Viscous and Heat-conductive
Gas
3.1. Initial Boundary Value Problems ....................... 143
3.1.1. Global Existence and Asymptotic Behavior of
Solutions ...................................... 143
3.1.2. Exponential Stability .......................... 153
3.1.3. Universal Attractors ........................... 154
3.2. The Cauchy Problem .................................... 154
3.2.1. Global Existence in H2(R) ...................... 154
3.2.2. Large-Time Behavior of Solutions ............... 159
3.3. Bibliographic Comments ................................ 164
4. A Polytropic Ideal Gas in Bounded Annular Domains in W1
4.1. Global Existence and Asymptotic Behavior in Я1 and
H2 .................................................... 167
4.1.1. Uniform A Priori Estimates in Hl ............... 175
4.1.2. Uniform a priori estimates in H2 ............... 187
4.1.3. Results in Eulerian Coordinates ................ 199
4.2. Exponential Stability in H4 ........................... 200
4.2.1. Main Results ................................... 200
4.2.2. Global Existence in H4 ......................... 202
4.2.3. A Nonlinear C0-Semigroup S(t) on H4 ............. 211
4.2.4. Exponential Stability in H4 .................... 222
4.3. Universal Attractors .................................. 227
4.3.1. Nonlinear Semigroups on H2 ..................... 230
4.3.2. Existence of an Absorbing Set in Hδ(1) .......... 231
4.3.3. Existence of an Absorbing Set in Hδ(2) .......... 236
4.3.4. Results of the Eulerian Coordinates ............ 241
4.3.5. Attractor in H4 ................................ 241
4.4. Bibliographic Comments ............................... 243
5. A Polytropic Viscous Gas with Cylinder Symmetry in  3
5.1. Main Results .......................................... 245
5.2. Global Existence and Exponential Stability in H1 ...... 249
5.3. Global Existence and Exponential Stability in H2 ...... 266
5.4. Global Existence and Exponential Stability in H4 ...... 268
5.4.1. Global Existence of Solutions in H4 ............ 268
5.4.2. Exponential Stability in H4+ .................... 285
5.5. Bibliographic Comments ................................ 290
6. One-dimensional Nonlinear Thermoviscoelasticity
6.1. Global Existence and Asymptotic Behavior of
Solutions ............................................. 293
6.2. Uniform A Priori Estimates ............................ 297
6.3. Exponential Stability and Maximal Attractors .......... 325
6.4. Exponential Stability in H1 and H2 .................... 331
6.5. Exponential Stability in H4 ........................... 332
6.6. Universal Attractors in H1 (i = 1,2,4) ................ 332
6.6.1. Existence of An Absorbing Set in Hδ1 ............ 332
6.6.2. Existence of An Absorbing Set in Hδ2 ............ 335
6.6.3. Existence of An Absorbing Set in Hδ4 ............ 336
6.7. Bibliographic Comments ................................ 336
7. A Nonlinear One-dimensional Thermoelastic System
with a Thermal Memory
7.1. Main Results .......................................... 339
7.2. Global Existence and Exponential Stability ............ 342
7.3. Bibliographic Comments ................................ 361
8. One-dimensional Thermoelastic Equations of Hyperbolic
Type
8.1. Global Existence ...................................... 363
8.2. Global Existence and Exponential Stability ............ 365
8.3. Bibliographic Comments ................................ 379
9. Blow-up for the Cauchy Problem in Nonlinear One-
dimensional Thermoelasticity
9.1. Introduction .......................................... 381
9.2. Main Results - Case I ................................. 382
9.3. Main Results - Case II ................................ 399
9.4. Bibliographic Comments ................................ 408
10.Large-Time Behavior of Energy in Multi-Dimensional
Elasticity
10.1.Polynomial Decay of Energy ............................ 409
10.1.1.Main Results ........................................ 411
10.1.2.Proof of Theorem 10.1.3 ............................. 413
10.2. Exponential Decay of Energy .......................... 421
10.2.1.Main Results ........................................ 421
10.2.2.Proof of Theorem 10.2.3 ............................. 426
10.3. Bibliographic Comments ............................... 433
Bibliography .................................................. 435
Index ......................................................... 463
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