| Preface | p. vii |
| Newtonian Filtration Equations | p. 1 |
| Introduction | p. 1 |
| Physical examples | p. 1 |
| Definitions of generalized solutions | p. 4 |
| Special solutions | p. 12 |
| Existence and Uniqueness of Solutions: One Dimensional Case | p. 13 |
| Uniqueness of solutions | p. 13 |
| Existence of solutions | p. 16 |
| Comparison theorems | p. 21 |
| Some extensions | p. 22 |
| Existence and Uniqueness of Solutions: Higher Dimensional Case | p. 26 |
| Comparison theorem and uniqueness of solutions | p. 26 |
| Existence of solutions | p. 31 |
| Some extensions | p. 34 |
| Regularity of Solutions: One Dimensional Case | p. 39 |
| Lemma | p. 39 |
| Regularity of solutions | p. 44 |
| Some extensions | p. 47 |
| Regularity of Solutions: Higher Dimensional Case | p. 48 |
| Generalized class B[subscript 2] | p. 49 |
| Some lemmas | p. 52 |
| Properties of functions in the generalized class B[subscript 2] | p. 55 |
| Holder continuity of solutions | p. 65 |
| Properties of the Free Boundary: One Dimensional Case | p. 71 |
| Finite propagation of disturbances | p. 71 |
| Localization and extinction of disturbances | p. 76 |
| Differential equation on the free boundary | p. 80 |
| Continuously differentiability of the free boundary | p. 83 |
| Some further results | p. 95 |
| Properties of the Free Boundary: Higher Dimensional Case | p. 96 |
| Monotonicity and Holder continuity of the free boundary | p. 96 |
| Lipschitz continuity of the free boundary | p. 114 |
| Differential equation on the free boundary | p. 122 |
| Initial Trace of Solutions | p. 123 |
| Harnack inequality | p. 124 |
| Main result | p. 132 |
| Extension of existence and uniqueness theorem | p. 138 |
| Other Problems | p. 138 |
| Equations with strongly nonlinear sources | p. 139 |
| Asymptotic properties of solutions | p. 143 |
| Non-Newtonian Filtration Equations | p. 147 |
| Introduction Preliminary Knowledge | p. 147 |
| Introduction Physical example | p. 147 |
| Basic spaces and some lemmas | p. 149 |
| Definitions of generalized solutions | p. 155 |
| Special solutions | p. 159 |
| Existence of Solutions | p. 160 |
| The case u[subscript 0] [set membership] C[superscript [infinity] subscript 0](R[superscript N]) or u[subscript 0] [set membership] L[superscript 1](R[superscript N]) [intersection of] L[superscript [infinity](R[superscript N]) | p. 161 |
| The case u[subscript 0] [set membership] L[superscript 1 subscript loc](R[superscript N]) | p. 175 |
| Some remarks | p. 188 |
| Harnack Inequality and the Initial Trace of Solutions | p. 189 |
| Local Harnack inequality | p. 189 |
| Global Harnack inequality | p. 196 |
| Initial trace of solutions | p. 201 |
| Regularity of Solutions | p. 204 |
| Boundedness of solutions | p. 204 |
| Boundedness of the gradient of solutions | p. 208 |
| Holder continuity of solutions | p. 212 |
| Holder continuity of the gradient of solutions | p. 215 |
| Uniqueness of Solutions | p. 230 |
| Auxiliary propositions | p. 230 |
| Uniqueness theorem and its proof | p. 246 |
| Properties of the Free Boundary | p. 253 |
| Monotonicity and Holder continuity of the free boundary | p. 253 |
| Lipschitz continuity of the free boundary | p. 262 |
| Other Problems | p. 263 |
| p-Laplacian equation with strongly nonlinear sources | p. 263 |
| Asymptotic properties of solutions | p. 266 |
| General Quasilinear Equations of Second Order | p. 269 |
| Introduction | p. 269 |
| Weakly Degenerate Equations in One Dimension | p. 271 |
| Uniqueness of bounded and measurable solutions | p. 272 |
| Existence of continuous solutions | p. 279 |
| Holder continuity of solutions | p. 284 |
| Some extensions | p. 286 |
| Weakly Degenerate Equations in Higher Dimension | p. 287 |
| Existence of continuous solutions for equations with two points of degeneracy | p. 289 |
| Uniqueness of BV solutions | p. 293 |
| Existence of BV solutions | p. 301 |
| Some extensions | p. 305 |
| Strongly Degenerate Equations in One Dimension | p. 306 |
| Definitions of solutions with discontinuity | p. 306 |
| Interior discontinuity condition | p. 308 |
| Uniqueness of BV solutions of the Cauchy problem | p. 314 |
| Formulation of the boundary value problem | p. 323 |
| Boundary discontinuity condition | p. 324 |
| Uniqueness of BV solutions of the first boundary value problem | p. 327 |
| Existence of BV solutions of the first boundary value problem | p. 328 |
| Some extensions | p. 332 |
| Equations with degeneracy at infinity | p. 333 |
| Properties of the curves of discontinuity | p. 334 |
| Degenerate Equations in Higher Dimension without Terms of Lower Order | p. 335 |
| Uniqueness of bounded and integrable solutions | p. 336 |
| A lemma on weak convergence | p. 341 |
| Existence of solutions | p. 344 |
| Finite propagation of disturbances | p. 349 |
| General Strongly Degenerate Equations in Higher Dimension | p. 353 |
| Existence of BV solutions | p. 354 |
| Some extensions | p. 362 |
| Appendix Classes BV and BV[subscript x] | p. 363 |
| Nonlinear Diffusion Equations of Higher Order | p. 369 |
| Introduction | p. 369 |
| Similarity Solutions of a Fourth Order Equation | p. 371 |
| Definition of similarity solutions | p. 372 |
| Existence and uniqueness of global solutions of the Cauchy problem | p. 373 |
| Regularity of solutions | p. 376 |
| Properties of solutions at zero points | p. 377 |
| Properties of unbounded solutions | p. 378 |
| Bounded solutions on the half line | p. 379 |
| Bounded solutions on the whole line | p. 386 |
| Properties of solutions in typical cases k = 1,2,3,4 | p. 388 |
| Behavior of similarity solutions as t [right arrow] 0[superscript +] | p. 397 |
| Equations with Double-Degeneracy | p. 399 |
| Existence of solutions | p. 399 |
| Uniqueness of solutions | p. 408 |
| Weighted energy equality of solutions | p. 416 |
| Some auxiliary inequalities | p. 417 |
| Finite propagation of disturbances | p. 418 |
| Asymptotic behavior of solutions | p. 421 |
| Extinction of solutions at finite time | p. 422 |
| Nonexistence of nonnegative solutions | p. 423 |
| Infinite propagation case | p. 424 |
| Cahn-Hilliard Equation with Constant Mobility | p. 425 |
| Existence of classical solutions | p. 425 |
| Blowing-up of solutions | p. 430 |
| Global existence of solutions for small initial value | p. 432 |
| Cahn-Hilliard Equations with Positive Concentration Dependent Mobility | p. 437 |
| A modified Campanato space | p. 438 |
| Holder norm estimates for a linear problem | p. 440 |
| Zero potential case | p. 448 |
| General case | p. 453 |
| Thin Film Equation | p. 454 |
| Definition of generalized solutions | p. 455 |
| Approximate solutions | p. 455 |
| Existence of solutions | p. 459 |
| Nonnegativity of solutions | p. 461 |
| Zeros of nonnegative solutions | p. 465 |
| Regularity of solutions | p. 466 |
| Monotonicity of the support of solutions | p. 468 |
| Cahn-Hilliard Equation with Degenerate Mobility | p. 470 |
| Models with degenerate mobility | p. 470 |
| Definition of physical solutions | p. 471 |
| Existence of solutions | p. 473 |
| Physical solutions | p. 474 |
| Bibliography | p. 479 |
| Table of Contents provided by Syndetics. All Rights Reserved. |
