Introduction
1 Part 1. Conservation of mass in Diffusion
1.1 Nonlinear diffusion, NLD
1.2 Outline and main results in standard NLD
Notations and comments
2 Study of the CM and its connections
2.1 CM and connections for the Heat Equation
2.2 Mass conservation and its connections for the PME
2.3 Extending the mass conservation law
2.4 Mass conservation and self-similarity for the PLE
3 Breakdown of the paradigm for very fast diffusion, VFDE
3.1 Extinction in the very fast diffusion range of the PME
3.2 The very fast diffusion range of the PLE
3.3 Positivity versus extinction in fast diffusion
3.4 Partial result for relative mass conservation
4 Mass conservation for the critical exponents
5 What happened to the fundamental solutions
5.1 The PME/FDE case
5.2 Large-time behaviour of general finite-mass solutions
5.3 The limit of the fundamental solution for the PLE
6 Non-conservation in some borderline cases
6.1 Logarithmic diffusion in \( N=2 \)
6.2 The PLE limit with \( p=1 \) in 1D. Total Variation Flow
6.3 Behaviour of general solutions
7 Part 2. Nonlinear fractional diffusion, NLFD
7.1 Mass conservation and its connections for the FHE
7.2 Outline of other results
8 Mass conservation in FPME
8.1 Mass conservation for the FPME with critical exponent
9 Conservation of mass for FPLE. Critical case
10 Part 3. Some related problems and open directions involving diffusion
10.1 The Doubly Nonlinear Equation
10.2 Anisotropic diffusion
10.3 Fractional PME with nonlocal pressure
10.4 Diffusion with convection. Fokker-Planck equations
10.5 Diffusion on manifolds
10.6 Other nonlinear diffusion equations
10.7 Other diffusive models
10.8 Other equations and systems