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

  • *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."

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  • 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.

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

    Citation:

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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} $

    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*} $

    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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