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. Hoelder 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. Hoelder 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
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\\} $$.- 11. Proof of Theorems 3.1 and 3.2.- 12. Proof of Theorem 4.1.- 13. Proof of Theorem 4.2.- 14. Proof of Theorem 4.3.- 15. Proof of Theorem 4.5.- 16. Proof of Theorems 5.1 and 5.2.- 17. Natural growth conditions.- 18. Bibliographical notes.- VI. Harnack estimates: the casep>2.- 1. Introduction.- 2. The intrinsic Harnack inequality.- 3. Local comparison functions.- 4. Proof of Theorem 2.1.- 5. Proof of Theorem 2.2.- 6. Global versus local estimates.- 7. Global Harnack estimates.- 8. Compactly supported initial data.- 9. Proof of Proposition 8.1.- 10. Proof of Proposition 8.1 continued.- 11. Proof of Proposition 8.1 concluded.- 12. The Cauchy problem with compactly supported initial data.- 13. Bibliographical notes.- VII. Harnack estimates and extinction profile for singular equations.- 1. The Harnack inequality.- 2. Extinction in finite time (bounded domains).- 3. Extinction in finite time (in RN).- 4. An integral Harnack inequality for all 1 2).- 4. Hoelder continuity ofDu (the case 1
2).- 5. Estimating the local average of |Dw| (the casep> 2).- 6. Estimating the local averages of w (the casep> 2).- 7. Comparing w and y (the case max $$ \\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.
