京都駅前セミナー＆大阪駅前セミナー

〜非線形現象の数理を考える〜

In this talk, we will focus on a fractional nonlinear diffusion equation posed on a bounded domain. This equation is a variant of the porous medium equation or the fast diffusion equation, where the Laplacian is replaced by a fractional Laplacian. In the porous medium case, solutions to this equation exhibit algebraic decay, whereas in the fast diffusion case, they undergo an extinction phenomenon: after a certain time, the solutions become uniformly zero.

Based on these estimates, we will study the large-time behavior of solutions and, in particular, show their convergence to separable variable solutions as time tends to infinity in the porous medium case or to the extinction time in the fast diffusion case. However, the extinction time is not analytically known, and to compute it, we will introduce a numerical scheme that preserves the qualitative properties of the continuous equation, namely algebraic decay and extinction. This scheme will allow us to numerically determine the rate of convergence to separable variable solutions, a question recently solved for the non-fractional fast diffusion case but still open in the fractional setting.