2015/08/31 by Matteo Bonforte, Bonforte, Matteo, Vázquez, Juan Luis · 1 citation
Computer Science · Mathematics · #35B45 #35B65 #35K55 #35K65 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1508.07871
openalex publication_date 2015/08/31 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28
We investigate quantitative properties of nonnegative solutions u(t,x)\≥ 0\nto the nonlinear fractional diffusion equation, \∂t u +\n\LF(u)=0 posed in a bounded domain, x\∈\Ω\⊂ \ℝN,\nwith appropriate homogeneous Dirichlet boundary conditions. As \L we\ncan use a quite general class of linear operators that includes the two most\ncommon versions of the fractional Laplacian (-\Δ)s, 0<s<1, in a\nbounded domain with zero Dirichlet boundary conditions, but it also includes\nmany other examples since our theory only needs some basic properties that are\ntypical of "linear heat semigroups." The nonlinearity F is assumed to be\nincreasing and is allowed to be degenerate, the prototype is the power case\nF(u)=|u|m-1u, with m>1.\n In this paper we propose a suitable class of solutions of the equation, and\ncover the basic theory: we prove existence, uniqueness of such solutions, and\nwe establish upper bounds of two forms (absolute bounds and smoothing effects),\nas well as weighted-L1 estimates. The class of solutions is very well suited\nfor that work. The standard Laplacian case s=1 is included and the linear\ncase m=1 can be recovered in the limit.\n In a companion paper [12], we will complete the study with more advanced\nestimates, like the upper and lower boundary behaviour and Harnack\ninequalities, for which the results of this paper are needed.\n