Geometric sturmian theory of nonlinear parabolic equations and applications

Victor A. Galaktionov1584884622, 9781584884620, 9780203998069

This book is a reference on second order parabolic partial differential equations that are used as models to help solve a broad class of engineering and physical problems. The analytical ideas used are geometric as opposed to super-sub solution methods that have limited application in the physical world. Sturm Theory is the cornerstone of this type of analysis, which was developed almost two hundred years ago, but then lost or forgotten. Recently the theory was revived and used to produce dramatic mathematical results. At present, there are a number of books that promote the Sturm Theory for ordinary differential equations, but none of them are devoted to parabolic partial differential equations.

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Table of contents :
GEOMETRIC STURMIAN THEORY of NONLINEAR PARABOLIC EQUATIONS and APPLICATIONS……Page 1
Contents……Page 4
Introduction: Sturm Theorems and Nonlinear Singular Parabolic Equations……Page 8
1.1 First Sturm Theorem: Nonincrease of the number of sign changes……Page 20
Results in the class of C1 functions……Page 22
Comments on Sturm’s evolution analysis of zeros……Page 24
Results in classes of finite regularity……Page 27
1.3 First aspects of intersection comparison of solutions of nonlinear parabolic equations……Page 28
1.4 Geometrically ordered flows: Transversality and concavity techniques……Page 30
1.5 Evolution B-equations preserving Sturmian properties……Page 33
Intersection comparison in the hodograph plane IR2……Page 34
Remarks and comments on the literature……Page 37
On spectra of multiple zeros in linear and quasilinear parabolic equations……Page 48
CHAPTER 2: Transversality, Concavity and Sign-Invariants. Solutions on Linear Invariant Subspaces……Page 54
2.1 Introduction: Filtration equation and concavity properties……Page 55
Transversality and concavity: finite propagation……Page 57
Concavity with infinite propagation……Page 64
Case of finite propagation……Page 66
On eventual concavity with infinite propagation……Page 70
Equations with absorption and source terms……Page 71
Equations with convection terms……Page 73
2.5 Singular equations with the p-Laplacian operator preserving concavity……Page 74
Three-dimensional set of explicit solutions on W3……Page 76
The choice of proper subsets of solutions……Page 77
B-concavity (convexity) is preserved in time……Page 78
Sign-invariants……Page 79
2.7 Various B-concavity properties for the porous medium equation and sign-invariants……Page 80
B-concavity with respect to fundamental solutions……Page 82
B-convexity to the subset of log Span……Page 83
2.9 B-concavity and transversality for the porous medium equation with source……Page 84
B-convexity on subspace of power functions……Page 87
B-convexity on subspace of hyperbolic functions……Page 88
The porous medium equation in IRN……Page 89
The fast diffusion equation in IRN……Page 93
Equation with the p-Laplace operator in IRN……Page 95
Linear sign-invariant associated with invariant subspace……Page 96
2.12 On general B-concavity via solutions on linear invariant subspaces……Page 97
Remarks and comments on the literature……Page 99
3.1 Introduction: Basic equations and concavity estimates……Page 104
3.2 Local concavity analysis via travelling wave solutions……Page 106
The set of travelling wave solutions……Page 107
Main result on concavity……Page 108
3.3 Concavity for the p-Laplacian equation with absorption……Page 112
Proper set of TW solutions……Page 113
B-concavity and sign-invariants……Page 115
Subset of similarity solutions is proper……Page 116
Semiconcavity estimate and sign-invariant……Page 117
3.6 B-concavity relative to incomplete functional subsets……Page 118
3.7 Eventual B-concavity……Page 119
Completeness and proper subsets……Page 120
Eventual B-concavity……Page 121
Remarks and comments on the literature……Page 122
4.1 Introduction: The blow-up problem……Page 124
4.2 Existence and nonexistence of singular blow-up travelling waves……Page 127
4.3 Discussion of the blow-up conditions. Pathological equations……Page 130
Nonexistence of nontrivial continuation……Page 132
Existence of nontrivial continuation……Page 135
4.5 The extinction problem……Page 138
Analysis of travelling wave solutions……Page 140
Extinction analysis……Page 142
Remarks and comments on the literature……Page 143
5.1 Introduction: First properties of incomplete blow-up……Page 144
5.2 Explicit proper blow-up travelling waves and first estimates of blow-up propagation……Page 146
5.3 Explicit blow-up solutions on an invariant subspace……Page 149
5.4 Lower speed estimate of blow-up interfaces……Page 152
5.5 Dynamical equation of blow-up interfaces……Page 153
Convexity……Page 154
Estimate of vxx from above……Page 155
Interface slope is finite and nondecreasing……Page 157
Interface equation……Page 158
Analytic continuation up to the blow-up time……Page 159
Analytic continuation up to the inflection point……Page 160
Breakdown of C2-regularity at inflection……Page 161
Extension to general solutions……Page 162
5.7 Large time behaviour of proper blow-up solutions……Page 163
5.8 Blow-up interfaces for the p-Laplacian equation with source……Page 164
Explicit parabolic solutions……Page 165
Linear TW solutions……Page 167
Linear explicit solutions……Page 168
Explicit blow-up solutions on an invariant set……Page 169
5.10 Examples of blow-up surfaces in IRN……Page 170
Nonsymmetric blow-up surfaces……Page 171
Explicit blow-up solutions on W……Page 173
Remarks and comments on the literature……Page 174
6.1 Introduction: The blow-up problem in IRN and critical exponents……Page 176
Order-preserving semigroups……Page 177
Extension of the semigroup……Page 179
6.3 Global continuation of nontrivial proper solutions……Page 181
6.4 On blow-up set in the limit case p = 2 – m……Page 182
6.5 Complete blow-up up to critical Sobolev exponent……Page 184
Subset of stationary solutions and the envelope……Page 185
Intersection comparison in radial geometry……Page 186
First result on complete blow-up……Page 187
Proof of complete blow-up: subcritical Sobolev range……Page 189
6.8 Complete blow-up of unfocused solutions……Page 191
Blow-up on a sphere……Page 192
6.9 Complete blow-up in the supercritical case……Page 193
Proof of the first theorem on complete blow-up……Page 195
6.10 Complete and incomplete blow-up for the equation with the p-Laplacian operator……Page 198
The limit case of incomplete blow-up……Page 199
6.11 Extinction problems in IRN and the criteria of complete and incomplete singularities……Page 200
Remarks and comments on the literature……Page 202
CHAPTER 7: Geometric Theory of Nonlinear Singular Parabolic Equations. Maximal Solutions……Page 206
7.1 Introduction: Main steps and concepts of the geometric theory……Page 207
Proper and improper TWs in one dimension……Page 210
Plane TWs for equations in IRN……Page 215
Pressure, interface operators, slopes and TW-diagram……Page 216
Gradient function……Page 219
Limit semigroups and maximal solutions……Page 220
Incomplete singularity and existence in 1D……Page 222
Existence for equations in IRN……Page 225
7.5 Complete singularities in IR and IRN. Infinite propagation and pathological equations……Page 226
Complete singularity (nonexistence) in 1D……Page 227
Nonexistence in IRN……Page 228
Infinite propagation and pathological PDEs……Page 229
Then the set B is complete…….Page 231
Sign-invariants……Page 232
B-number……Page 234
Eventual B-concavity……Page 235
Strong Maximum Principle for interfaces……Page 236
B-classes, transversality and gradient estimates……Page 237
Instantaneous smoothing phenomenon in B-classes……Page 239
Lipschitz continuity of interfaces and level propagation……Page 242
Optimal moduli of continuity in x and t……Page 243
Eventual smoothing and waiting time phenomena……Page 244
7.8 Transversality and smoothing in the radial geometry in IRN……Page 246
7.9 B-concavity in the radial geometry in IRN……Page 249
7.10 Interface operators and equations, uniqueness……Page 250
The case 0 = IR……Page 251
Interfaces in the case 0 6= IR……Page 254
On interface velocity estimates in IRN……Page 255
Uniqueness for FBPs for maximal solutions……Page 256
7.11 Applications to various nonlinear models with extinction and blow-up singularities in IR and IRN……Page 257
Quasilinear heat equations with absorption……Page 258
Blow-up interfaces for quasilinear equation with source……Page 265
The dual PME with absorption……Page 266
Blow-up for the dual PME with source……Page 268
General quasilinear heat equation with absorption……Page 269
Applications to equations from mean curvature flows……Page 272
On a generalization with discontinuous limit semigroup……Page 273
Fully nonlinear equation from detonation theory……Page 274
Remarks and comments on the literature……Page 276
8.1 Introduction: One-phase free-boundary Stefan and Florin problems……Page 280
8.2 Classification of free-boundary problems for the heat equation……Page 284
8.3 Classification of free-boundary problems for the quadratic porous medium equation……Page 288
Classification of proper FBPs……Page 289
8.4 On general one-phase free-boundary problems……Page 291
8.5 Higher-order free-boundary problems for the porous medium equation with absorption……Page 293
8.6 Higher-order free-boundary problems for the dual porous medium equation with singular absorption……Page 296
Two-phase FBPs for the heat equation……Page 297
FBPs for the sign PME with absorption……Page 299
Remarks and comments on the literature……Page 300
9.1 Introduction: Solutions of changing sign and the phenomenon of singular propagation……Page 302
9.2 Application: the sign porous medium equation with singular absorption……Page 308
On interior gradient blow-up of bounded solutions……Page 310
9.3 On propagation of singularity curves……Page 311
Remarks and comments on the literature……Page 313
10.1 Introduction: New nonlinear models with discontinuous semigroups……Page 314
10.2 Existence and nonexistence results for the hydrodynamic version……Page 315
Subset of travelling waves……Page 316
Discontinuity: first example of complete singularity……Page 317
Nonexistence for solutions changing sign……Page 318
Positivity……Page 319
Continuity: local comparison with similarity solutions……Page 321
10.3 A generalized model with complete and incomplete singularities……Page 323
Positivity and finite propagation……Page 324
10.4 Complete singularity in the Cauchy problem for the Zhang equation……Page 325
Existence for bounded initial data……Page 326
A priori bound……Page 327
Self-similar solutions: local singularity formation……Page 328
Instant shape simplification of initial data……Page 329
Generalized models……Page 332
10.6 Discontinuous limit semigroups and operator of shape simplification for singular equations in IRN……Page 334
Remarks and comments on the literature……Page 335
11.1 Equations in IRN with blow-up and spatial singularities: discontinuous semigroups and singular initial layers……Page 336
Critical non-autonomous singularity for the PME with source……Page 337
On oscillatory solutions of changing sign……Page 342
Examples of incomplete critical singularity……Page 344
Other examples of critical complete and incomplete blow-up……Page 345
On local non-solvability of critical stationary equations……Page 346
11.2 When do singular interfaces not move?……Page 347
One-dimensional problems……Page 348
Non-moving singular interfaces in IRN……Page 350
Remarks and comments on the literature……Page 351
On limit minimal semigroups for singular initial data……Page 353
References……Page 356
List of Frequently Used Abbreviations……Page 376