# Monotone iterations method for fractional diffusion equations

### Abstract

In recent years, there has been a growing interest on non-local

models because of their relevance in many practical applications. A

widely studied class of non-local models involves fractional order

operators. They usually describe anomalous diffusion. In

particular, these equations provide a more faithful representation

of the long-memory and nonlocal dependence of diffusion in fractal

and porous media, heat flow in media with memory, dynamics of

protein in cells etc.

For $a\in (0, 1)$, we investigate the nonautonomous fractional

diffusion equation:

$D^a_{*,t} u - Au = f(x, t,u),$

where

$D^a_{*,t}$ is the Caputo fractional derivative and $A$ is a

uniformly elliptic operator with smooth coefficients depending on

space and time. We consider these equations together with initial

and quasilinear boundary conditions.

The solvability of such problems in H\"older spaces presupposes

rigid restrictions on the given initial data. These compatibility

conditions have no physical meaning and, therefore, they can be

avoided, if the solution is sought in larger spaces, for instance in

weighted H\"older spaces.

We give general existence and uniqueness result and

provide some examples of applications of the main theorem. The main

tool is the monotone iterations method. Preliminary we developed the

linear theory with existence and comparison results. The principle

use of the positivity lemma is the construction of a monotone

sequences for our problem. Initial iteration may be taken as either

an upper solution or a lower solution. We provide some examples of

upper and lower solution for the case of linear equations and

quasilinear boundary conditions. We notice that this approach can

also be extended to other problems and systems of fractional

equations as soon as we will be able to construct appropriate upper

and lower solutions.

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