Degenerate Parabolic Equations

I. Notation and function spaces.- §1. Some notation.- §2. Basic facts aboutW1,p(?) andWo1,p(?).- §3. Parabolic spaces and embeddings.- §4. Auxiliary lemmas.- §5. Bibliographical notes.- II. Weak solutions and local energy estimates.- §1. Quasilinear degenerate or singular equations.- §2. Boundary value problems.- §3. Local integral inequalities.- §4. Energy estimates near the boundary.- §5. Restricted structures: the levelskand the constant ?.- §6. Bibliographical notes.- III. Hölder continuity of solutions of degenerate parabolic equations.- §1. The regularity theorem.- §2. Preliminaries.- §3. The main proposition.- §4. The first alternative.- §5. The first alternative continued.- §6. The first alternative concluded.- §7. The second alternative.- §8. The second alternative continued.- §9. The second alternative concluded.- §10. Proof of Proposition 3.1.- §11. Regularity up tot= 0.- §12. Regularity up toST. Dirichlet data.- §13. Regularity atST. Variational data.- §14. Remarks on stability.- §15. Bibliographical notes.- IV. Hölder continuity of solutions of singular parabolic equations.- §1. Singular equations and the regularity theorems.- §2. The main proposition.- §3. Preliminaries.- §4. Rescaled iterations.- §5. The first alternative.- §6. Proof of Lemma 5.1. Integral inequalities.- §7. An auxiliary proposition.- §8. Proof of Proposition 7.1 when (7.6) holds.- §9. Removing the assumption (6.1).- §10. The second alternative.- §11. The second alternative concluded.- §12. Proof of the main proposition.- §13. Boundary regularity.- §14. Miscellaneous remarks.- §15. Bibliographical notes.- V. Boundedness of weak solutions.- §1. Introduction.- §2. Quasilinear parabolic equations.- §3. Sup-bounds.- §4. Homogeneous structures. 2.- §5. Homogeneous structures. The singular case 1 <p< 2.- §6. Energy estimates.- §7. Local iterative inequalities.- §8. Local iterative inequalities $$
\left( {p > max\left\{ {1;\frac{{2N}}
{{N + 2}}} \right\}} \right)
$$.- §9. Global iterative inequalities.- §10. Homogeneous structures and $$
1 < p \leqslant max\left\{ {1;\frac{{2N}}
{{N + 2}}} \right\}
$$.- §11. Proof of Theorems 3.1 and 3.2.- §12. Proof of Theorem 4.1.- §13. Proof of Theorem 4.2.- §14. Proof of Theorem 4.3.- §15. Proof of Theorem 4.5.- §16. Proof of Theorems 5.1 and 5.2.- §17. Natural growth conditions.- §18. Bibliographical notes.- VI. Harnack estimates: the casep>2.- §1. Introduction.- §2. The intrinsic Harnack inequality.- §3. Local comparison functions.- §4. Proof of Theorem 2.1.- §5. Proof of Theorem 2.2.- §6. Global versus local estimates.- §7. Global Harnack estimates.- §8. Compactly supported initial data.- §9. Proof of Proposition 8.1.- §10. Proof of Proposition 8.1 continued.- §11. Proof of Proposition 8.1 concluded.- §12. The Cauchy problem with compactly supported initial data.- §13. Bibliographical notes.- VII. Harnack estimates and extinction profile for singular equations.- §1. The Harnack inequality.- §2. Extinction in finite time (bounded domains).- §3. Extinction in finite time (in RN).- §4. An integral Harnack inequality for all 1 2).- §4. Hölder continuity ofDu (the case 1 <p< 2).- §5. Some algebraic Lemmas.- §6. Linear parabolic systems with constant coefficients.- §7. The perturbation lemma.- §8. Proof of Proposition 1.1-(i).- §9. Proof of Proposition 1.1-(ii).- §10. Proof of Proposition 1.1-(iii).- §11. Proof of Proposition 1.1 concluded.- §12. Proof of Proposition 1.2-(i).- §13. Proof of Proposition 1.2 concluded.- §14. General structures.- §15. Bibliographical notes.- X. Parabolicp-systems: boundary regularity.- §1. Introduction.- §2. Flattening the boundary.- §3. An iteration lemma.- §4. Comparing w and y (the casep> 2).- §5. Estimating the local average of |Dw| (the casep> 2).- §6. Estimating the local averages of w (the casep> 2).- §7. Comparing w and y (the case max $$
\left\{ {1;\tfrac{{2N}}
{{N + 2}}} \right\} < p < 2
$$).- §8. Estimating the local average of |Dw|.- §9. Bibliographical notes.- XI. Non-negative solutions in ?T. The casep>2.- §1. Introduction.- §2. Behaviour of non-negative solutions as |x| ? ? and as t ? 0.- §3. Proof of (2.4).- §4. Initial traces.- §5. Estimating |Du|p?1 in ?T.- §6. Uniqueness for data inLloc1(RN).- §7. Solving the Cauchy problem.- §8. Bibliographical notes.- XII. Non-negative solutions in ?T. The case 1 The uniqueness theorem.- §6. An auxiliary proposition.- §7. Proof of the uniqueness theorem.- §8. Solving the Cauchy problem.- §9. Compactness in the space variables.- §10. Compactness in thetvariable.- §11. More on the time—compactness.- §12. The limiting process.- §13. Bounded solutions. A counterexample.- §14. Bibliographical notes.