Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications (Chapman & Hall/CRC Applied Mathematics & Nonlinear Science Book 3)

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. 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

Unlike the classical Sturm theorems on the zeros of solutions of second-order ODEs, Sturm's evolution zero set analysis for parabolic PDEs did not attract much attention in the 19th century, and, in fact, it was lost or forgotten for almost a century. Briefly revived by Pólya in the 1930's and rediscovered in part several times since, it was not until the 1980's that the Sturmian argument for PDEs began to penetrate into the theory of parabolic equations and was found to have several fundamental applications.

Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications focuses on geometric aspects of the intersection comparison for nonlinear models creating finite-time singularities. After introducing the original Sturm zero set results for linear parabolic equations and the basic concepts of geometric analysis, the author presents the main concepts and regularity results of the geometric intersection theory (G-theory). Here he considers the general singular equation and presents the geometric notions related to the regularity and interface propagation of solutions. In the general setting, the author describes the main aspects of the ODE-PDE duality, proves existence and nonexistence theorems, establishes uniqueness and optimal Bernstein-type estimates, and derives interface equations, including higher-order equations. The final two chapters explore some special aspects of discontinuous and continuous limit semigroups generated by singular parabolic equations.

Much of the information presented here has never before been published in book form. Readable and self-contained, this book forms a unique and outstanding reference on second-order parabolic PDEs used as models for a wide range of physical problems.

"Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications focuses on geometric aspects of the intersection comparison for nonlinear models creating finite-time singularities. After introducing the original Sturm zero set results for linear parabolic equations and the basic concepts of geometric analysis, the author presents the main concepts and regularity results of the geometric intersection theory (G-theory). Here he considers the general singular equation and presents the geometric notions related to the regularity and interface propagation of solutions. In the general setting, the author describes the main aspects of the ODE-PDE duality, proves existence and nonexistence theorems, establishes uniqueness and optimal Bernstein-type estimates, and derives interface equations, including higher-order equations. The final two chapters explore some special aspects of discontinuous and continuous limit semigroups generated by singular parabolic equations." "Much of the information presented here has never before been published in book form or even in mathematics journals. This book forms a unique reference on second-order parabolic PDEs used as models for a wide range of physical problems."--BOOK JACKET. Introduction: Sturm Theorems and Nonlinear Singular Parabolic EquationsSturm Theorems for Linear Parabolic Equations and Intersection Comparison. B-equationsFirst Sturm Theorem: Nonincrease of the number of sign changesSecond Sturm Theorem: Evolution formation and collapse of multiple zerosFirst aspects of intersection comparison of solutions of nonlinear parabolic equationsGeometrically ordered flows: Transversality and concavity techniquesEvolution B-equations preserving Sturmian properties Transversality, Concavity and Sign-Invariants. Solutions on Linear Invariant SubspacesIntroduction: Fi Cover; Title; Copyright; Contents; Introduction: Sturm Theorems and Nonlinear Singular Parabolic Equations; CHAPTER 1: Sturm Theorems for Linear Parabolic Equations and Intersection Comparison. B-equations; CHAPTER 2: Transversality, Concavity and Sign-Invariants. Solutions on Linear Invariant Subspaces; CHAPTER 3: B-Concavity and Transversality on Nonlinear Subsets for Quasilinear Heat Equations; CHAPTER 4: Eventual B-convexity: a Criterion of Complete Blow-up and Extinction for Quasilinear Heat Equations; CHAPTER 5: Blow-up Interfaces for Quasilinear Heat Equations CHAPTER 6: Complete and Incomplete Blow-up in Several Space DimensionsCHAPTER 7: Geometric Theory of Nonlinear Singular Parabolic Equations. Maximal Solutions; CHAPTER 8: Geometric Theory of Generalized Free-Boundary Problems. Non-Maximal Solutions; CHAPTER 9: Regularity of Solutions of Changing Sign; CHAPTER 10: Discontinuous Limit Semigroups for the Singular Zhang Equation; CHAPTER 11: Further Examples of Discontinuous and Continuous Limit Semigroups; References; List of Frequently Used Abbreviations; Index