Contents
Preface xi
Chapter 1. The Cauchy problem
for linear nonlocal diffusion 1
1.1. The Cauchy problem 1
1.1.1. Existence and uniqueness 5
1.1.2. Asymptotic behaviour 6
1.2. Refined asymptotics 10
1.2.1. Estimates on the regular part of the fundamental solution 12
1.2.2. Asymptotics for the higher order terms 17
1.2.3. A different approach 20
1.3. Rescaling the kernel. A nonlocal approximation
of the heat equation 22
1.4. Higher order problems 23
1.4.1. Existence and uniqueness 24
1.4.2. Asymptotic behaviour 25
1.4.3. Rescaling the kernel in a higher order problem 28
Bibliographical notes 29
Chapter 2. The Dirichlet problem
for linear nonlocal diffusion 31
2.1. The homogeneous Dirichlet problem 31
2.1.1. Asymptotic behaviour 32
2.2. The nonhomogeneous Dirichlet problem 36
2.2.1. Existence, uniqueness and a comparison principle 36
2.2.2. Convergence to the heat equation when rescaling the kernel 38
Bibliographical notes 40
Chapter 3. The Neumann problem
for linear nonlocal diffusion 41
3.1. The homogeneous Neumann problem 41
3.1.1. Asymptotic behaviour 42
3.2. The nonhomogeneous Neumann problem 45
3.2.1. Existence and uniqueness 46
3.2.2. Rescaling the kernels. Convergence to the heat equation 48
3.2.3. Uniform convergence in the homogeneous case 54
3.2.4. An
L1-convergence
result in the nonhomogeneous case 56
3.2.5. A weak convergence result in the nonhomogeneous case 57
Bibliographical notes 63
vii
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