Communications in Mathematical Sciences

Volume 20 (2022)

Number 5

Initial mixed-boundary value problem for anisotropic fractional degenerate parabolic equations

Pages: 1279 – 1304



Gerardo Huaroto (Instituto de Matemática, Universidade Federal de Alagoas, Maceio, Alagoas, Brazil)

Wladimir Neves (Instituto de Matemática, Universidade Federal do Rio de Janeiro, RJ, Brazil )


We consider an initial mixed-boundary value problem for anisotropic fractional type degenerate parabolic equations posed in bounded domains. Namely, we consider that the boundary of the domain splits into two parts. In one of them, we impose a Dirichlet boundary condition and in the other part a Neumann condition. Under this mixed-boundary condition, we show the existence of solutions for measurable and bounded non-negative initial data. The nonlocal anisotropic diffusion effect relies on an inverse of a $s$-fractional type elliptic operator, and the solvability is proved for any $s \in (0,1)$.


fractional elliptic operator, initial mixed-boundary value problem, Dirichlet–Neumann homogeneous boundary condition, anisotropic problem

2010 Mathematics Subject Classification

35D30, 35K55, 35K61, 35K65

Received 3 May 2021

Received revised 21 November 2021

Accepted 21 November 2021

Published 26 May 2022