Nonlinear evolution equations in Banach spaces [preprint]

Title page......Page 1
1.1 Operators......Page 7
1.2 Classical and Strong Solutions......Page 8
1.3 Mild Solutions......Page 9
1.4 Mild Versus Strong......Page 11
1.5 Further Properties of Mild Solutions......Page 16
1.6 Semigroups and Generators......Page 22
1.7 Exercises......Page 23
2.1 Definition and Examples of Accretive Operators......Page 28
2.2 The Bracket......Page 31
2.3 The Duality Map......Page 35
2.4 The Bracket, the Duality Map and Accretivity......Page 36
2.5 Closure and the Lim Inf......Page 38
2.6 Sums of Accretive Operators and s-accretivity......Page 39
2.7 Exercises......Page 41
3.1 Existence and Uniqueness of Solutions - Statement of Results......Page 48
3.2 Solvability of General Discretizations......Page 50
3.3 The Main Estimates - Proofs......Page 51
3.4 Existence, Uniqueness and Continuity - Proofs......Page 55
3.5 Semigroups Governed by Accretive Operators......Page 57
3.6 Exercises......Page 59
Chapter 4 Resolvents, the Exponential Formula and Mild Solutions of u' + Au \ni f......Page 65
4.1 The Range Condition and the Exponential Formula......Page 66
4.2 Properties of the Resolvent......Page 67
4.3 The Inhomogeneous Equation......Page 70
4.4 Exercises......Page 73
5.2 Mild Solutions of Linear Equations......Page 77
5.3 Generation of Semi groups of Bounded Linear Operators......Page 79
5.4 Variation of Parameters and the Inhomogeneous Equation......Page 86
5.5 Exercises......Page 88
6.1 Integral Solutions......Page 92
6.2 Integral and Mild Solutions......Page 95
6.3 Exercises......Page 102
Chapter 7 Strong Solutions and Regularity of Mild Solutions......Page 108
7.1 Pointwise Derivatives of Mild Solutions......Page 109
7.2 Lipschitz Continuity and the Radon-Nikodym Property......Page 111
7.3 Differentiability of Sohitions of u' + Au \ni 0......Page 113
7.4 Refinements Under Convexity Conditions on X......Page 115
7.5 Exercises......Page 117
8.1 m-Accretive Operators......Page 122
8.2 Maximal Monotone Graphs in R and Subdifferentials in Hilbert Spaces......Page 123
8.3 Properties of m-Accretive Operators and the Yosida Approximation......Page 126
8.4 Exercises......Page 128
9.1 Translation Semigroups......Page 135
9.2 The Scalar Conservation Law......Page 140
9.2.1 Comparison of Notions of Solutions of the Conservation Law......Page 144
9.2.2 A Generalized Divergence......Page 150
9.3 Hamilton-Jacobi Equations......Page 152
9.3.1 Viscosity Solutions......Page 156
9.3.2 Proofs of Propositions 9.22 and 9.23......Page 159
9.3.3 The Hamilton-Jacobi Semigroup......Page 161
9.4 Exercises......Page 162
Chapter 10 m-Accretive Differential Operators of Second Order......Page 163
11.1 Convergence of Operators and Dependence on A......Page 164
11.2 An Application to Yosida Approximations......Page 171
11.3 Exercises......Page 172
12.1 A Generalization of the Exponential Formula......Page 177
12.2 Product Formulas......Page 181
12.2 Exercises......Page 185
13.1 The Main Results......Page 191
13.2 A Reduction to the Inhomogeneous Case......Page 192
13.3 A Linear Approximation Result......Page 196
13.4 Proofs of the Main Results......Page 198
13.5 Exercises......Page 199
14.1 Definition and Elementary Properties of the Generalized Domain D(A)......Page 200
14.2 D(A) and Lipschitz Continuity......Page 201
14.3 Interpretations of D(A) in X......Page 207
14.4 Exercises......Page 212
15.1 A Necessary and Sufficient Condition......Page 213
15.2 Tangency Conditions......Page 214
15.3 Proof of Theorem 15.1......Page 217
15.4 Exercises......Page 222
16.1 Relatively Continuous Perturbations......Page 223
16.2 A Characterization and Applications to Continuous Perturbations......Page 225
16.3 Perturbations in Uniformly Smooth Spaces......Page 227
16.4 A + B is Continuous in B......Page 230
16.5 Exercises......Page 231
Chapter 17 Compactness......Page 234
17.1 Review of Compactness......Page 235
17.2 Compact Semigroups......Page 236
17.3 Compactness of'llajectories in the Inhomogeneous Problem......Page 237
17.4 Compactness of the Evolution Operator......Page 239
17.5 Exercises......Page 243
18.1 A Summary of the Main Results......Page 244
18.2 Some Technical Le mmas......Page 247
18.3 Proofs of the Main ltesults......Page 251
18.5 Exercises......Page 261
19.1 Liapunov Functions......Page 262
19.2 Liapunov Couples and Sequences......Page 265
19.3 Convex Liapunov Functionals......Page 268
19.4 Order-Preservation and T-accretivity......Page 272
19.5 Exercises......Page 277
Appendice......Page 279