The Cauchy-Dirichlet problem for singular nonlocal diffusions on bounded domains

Matteo Bonforte; Peio Ibarrondo; Mikel Ispizua

doi:10.3934/dcds.2022112

Article Contents

2023, Volume 43, Issue 3&4: 1090-1142. Doi: 10.3934/dcds.2022112

This issue Previous Article Constructive stability results in interpolation inequalities and explicit improvements of decay rates of fast diffusion equations Next Article Cauchy-Dirichlet problems for the porous medium equation

The Cauchy-Dirichlet problem for singular nonlocal diffusions on bounded domains

1.
Departamento de Matemáticas, Universidad Autónoma de Madrid, ICMAT - Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM, Campus de Cantoblanco, 28049 Madrid, Spain
2.
Departamento de Matemáticas, Universidad Autónoma de Madrid, Campus de Cantoblanco, 28049 Madrid, Spain

^*Corresponding author: Matteo Bonforte
^*Corresponding author: Matteo Bonforte

Dedicated to Juan Luis Vázquez for his 75th birthday. "Con ammirazione ed affetto per Juan Luis, Maestro della diffusione nonlineare."

Received: February 2022

Revised: March 2022

Early access: August 2022

Published: March 2023

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Abstract

We study the Cauchy-Dirichlet Problem (CDP) for a nonlinear and nonlocal diffusion equation of singular type of the form $ \partial_t u = - \mathcal{L} u^m $ posed on a bounded Euclidean domain $ \Omega\subset \mathbb{R}^N $ with smooth boundary and $ N\ge 1 $. The linear diffusion operator $ \mathcal{L} $ is a sub-Markovian operator, allowed to be of nonlocal type, while the nonlinearity is of singular type, namely $ u^m = |u|^{m-1}u $ with $ 0<m<1 $. The prototype equation is the Fractional Fast Diffusion Equation (FFDE), when $ \mathcal{L} $ is one of the three possible Dirichlet Fractional Laplacians on $ \Omega $.

We provide a complete basic theory for solutions to (CDP): existence and uniqueness in the biggest class of data known so far, both for nonnegative and signed solutions; sharp smoothing estimates: classical $ L^p-L^\infty $ smoothing effects, and new weighted estimates, which represent a novelty also in local case, i.e. $ u_t = \Delta u^m $. We compare two strategies to prove smoothing effects: Moser iteration VS Green function method.

Due to the singular nonlinearity and to presence of nonlocal diffusion operators, the question of how solutions satisfy the lateral boundary conditions is delicate and we answer it by quantitative upper boundary estimates.

Finally, we show that solutions extinguish in finite time and we provide upper and lower estimates for the extinction time, together with explicit sharp extinction rates in different norms.

Keywords:

Mathematics Subject Classification: Primary: 5K55, 35B65, 35A01, 35B45, 35R11; Secondary: 35K67, 35K61, 35A02.

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Figure 1. $ L^p-L^\infty $ and $ H^*-L^\infty $ smoothing effects in the different fast diffusion regimes in relation with the critical exponents: $\mathbf{p_c} = \frac{N(1-m)}{2s}\qquad\mathbf{m_c} = \frac{N-2s}{N}\qquad\mathbf{m_s} = \frac{N-2s}{N+2s} $

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Figure 2. Weighted and unweighted smoothing effects in the different fast diffusion regimes in relation with the critical exponents: $ \begin{gather*} \mathbf{m_c} = \frac{N-2s}{N} \quad \mathbf{m_{c, \gamma }} = \frac{N+\gamma-2s}{N} \quad \mathbf{m_s} = \frac{N-2s}{N+2s} \quad \mathbf{p_c} = \frac{N(1-m)}{2s} \quad \mathbf{p_{c, \gamma }} = \frac{N(1-m)}{2s-\gamma} \end{gather*} $

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Figure 3. On the left side, the green line in the $ (m, p)- $plane. On the right side, the weighted green line in the $ (m, p)- $plane

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