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.

Keywords

• fractional; holder spaces; large-time behaviour; comparison principle