Preface 1. Elliptic equations: Hamack estimates and Holder continuity 2. Parabolic equations: Hamack estimates and Holder continuity 3. Parabolic equations and systems 4. Main results Ⅰ. Notation and function spaces 1. Some notation 2. Basic facts about W1,and W2 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 levels k and the constant 6. Bibliographical notes Ⅲ. Holder continuity of solutions of degenerate parabolic equations 1. The regularity theorem 2. Preliminaries 3. The main proposition 4. The first alternative 5. The first altemative 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 to t = 0 12. Regularity up to ST. Dirichlet data 13. Regularity at ST. Variational data 14. Remarks on stability 15. Bibliographical notes Ⅳ. Holder 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. The degenerate case p > 2 5. Homogeneous structures. The singular case 1 < p < 2 6. Energy estimates 7. Local iterative inequalities 8. Local iterative inequalities 9. Global iterative inequalities 10. Homogeneous structures and 1 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. i concluded 12. The Canchy problem with compactly supported initial data 13. Bibliographical notes Ⅶ. Harnack estimates and extinction profile for singular equations 1. The Hamack inequality 2. Extinction in finite time (bounded domains) 3. Extinction in finite time (in RN) 4. An integral Harnack inequality for all 1 < p < 2 5. Sup-estimates for < p < 2 6. Local subsolutions 7. Time expansion of positivity 8. Space-time configurations 9. Proof of the Harnack inequality 10. Proof of Theorem 1.2 11. Bibliographical notes Ⅷ. Degenerate and singular parabolic systems 1. Introduction 2. Bonndedness of weak solutions 3. Weak differentiability of |Du|: Du and energy estimates for |Du| 4. Bonndedness of |Du|. Qualitative estimates 5. Quantitative sup-bounds of |Du| 6. General structures 7. Bibliographical notes Ⅸ. Parabolic p-systems: Holder continuity of Du 1. The main theorem 2. Estimating the oscillation of Du 3. Holder continuity of Du (the case p > 2 ) 4. Holder continuity of Du (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 Parabolic p-systems: boundary regularity 1. Introduction 2. Flattening the boundary 3. An iteration lemma 4. Comparing w and v (the case p > 2) 5. Estimating the local average of Ho (thecase p > 2) 6. Estimating the local averages of w (the case p > 2) 7. Comparing w and v 8. Estimating the local average of |Dw| 9. Bibfiographieal notes Ⅺ. Non-negative solutions in ∑T. The case p>2 1. Introduction 2. Behaviour of non-negative solutions as |x| 3. Proof of (2.4) 4. Initial traces 5. Estimating 6. Uniqueness for data in L(RN) 7. Solving the Cauchy problem 8. Bibliographical notes Ⅻ. Non-negative solutions in ∑r. The case 1

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