Monotone iterations method for fractional diffusion equations

Анотація

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.