Uniqueness and properties of distributional solutions of nonlocal equations of porous medium type

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• 作者：Fé ; lix del Teso^a ; ^{felix.delteso@uam.es" class="auth_mail" title="E-mail the corresponding author} ; Jø ; rgen Endal^b ; ^{jorgen.endal@math.ntnu.no" class="auth_mail" title="E-mail the corresponding author} ; Espen R. Jakobsen^b ; ^{erj@math.ntnu.no" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35A02 ; 35B30 ; 35B35 ; 35B53 ; 35D30 ; 35J15 ; 35K59 ; 35K65 ; 35L65 ; 35R09 ; 35R11
• 刊名：Advances in Mathematics
• 出版年：2017
• 出版时间：10 January 2017
• 年：2017
• 卷：305
• 期：Complete
• 页码：78-143
• 全文大小：985 K

文摘

We study the uniqueness, existence, and properties of bounded distributional solutions of the initial value problem for the anomalous diffusion equation ∂_tu−L^μ[φ(u)]=0${\partial }_{t}u-{\mathcal{L}}^{\mu }[\phi (u)]=0$. Here L^μ${\mathcal{L}}^{\mu }$ can be any nonlocal symmetric degenerate elliptic operator including the fractional Laplacian and numerical discretizations of this operator. The function φ:R→R$\phi :\mathbb{R}\to \mathbb{R}$ is only assumed to be continuous and nondecreasing. The class of equations include nonlocal (generalized) porous medium equations, fast diffusion equations, and Stefan problems. In addition to very general uniqueness and existence results, we obtain stability, L¹$L^1$-contraction, and a priori estimates. We also study local limits, continuous dependence, and properties and convergence of a numerical approximation of our equations.