| Abstract: |
| This talk is concerned with the Cauchy-Dirichlet problem for a fractional nonlinear diffusion equation posed on a bounded domain. It is well-known that solutions to the standard (non-fractional) fast diffusion equation extinct in finite time, and that rescaled solution converges to asymptotic profiles which are solutions to the Dirichlet problem of a nonlinear elliptic equation. First, we introduce an extension of this result in the fractional case, using an energy method. It is based on precise energy decay estimates, but is only available when the exponent in the nonlinearity is below the Sobolev critical exponent. Then, we introduce a numerical scheme for the simulation of the fractional fast diffusion equation. The scheme is based on a discrete fractional Laplacian recently introduced by Huang and Oberman, which admits a variational structure very similar to the continuous fractional Laplacian, and it allows us to extend energy decay estimates to the numerical scheme. We use this scheme to qualitatively answer questions which are out of the frame of our theoretical result, such as precise estimation of the extinction, time, and convergence to asymptotic profiles when the exponent in the nonlinearity is in the Sobolev super-critical range. |
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