Linear and nonlinear diffusion problems

Introduction......Page 7
Sectorial operators......Page 9
The second order derivative in the real line......Page 21
The operator Au=u'' in a bounded interval, with Dirichlet boundary conditions......Page 23
The Laplacian in RN......Page 25
Some abstract examples......Page 27
The Dirichlet Laplacian in a bounded open set......Page 29
More general operators......Page 31
The interpolation spaces DA(,)......Page 35
Strict, classical, and mild solutions......Page 43
Behavior of etA......Page 53
Behavior of etA for a hyperbolic A......Page 54
Bounded solutions in [0,+ )......Page 58
Bounded solutions in (- ,0]......Page 60
Bounded solutions in R......Page 62
Solutions with exponential growth and exponential decay......Page 64
Local existence, uniqueness, regularity......Page 69
The maximally defined solution......Page 71
Reaction -- diffusion equations and systems......Page 73
The maximum principle......Page 75
Linearized stability......Page 81
Linearized instability......Page 83
The saddle point property......Page 84
A Cauchy-Dirichlet problem......Page 87
A Cauchy-Neumann problem......Page 88
Vector-valued integration......Page 91
Basic Spectral Theory......Page 97