Monotone iterations method for fractional diffusion equations

Abstract

In recent years, there has been a growing interest on non-localmodels because of their relevance in many practical applications. Awidely studied class of non-local models involves fractional orderoperators. They usually describe anomalous diffusion. Inparticular, these equations provide a more faithful representationof the long-memory and nonlocal dependence of diffusion in fractaland porous media, heat flow in media with memory, dynamics ofprotein in cells etc. For $a\in (0, 1)$, we investigate the nonautonomous fractionaldiffusion equation: $D^a_{*,t} u - Au = f(x, t,u),$  where$D^a_{*,t}$ is the Caputo fractional derivative and $A$ is auniformly elliptic operator with smooth coefficients depending onspace and time. We consider these equations together with initialand quasilinear boundary conditions. The solvability of such problems in H\"older spaces presupposesrigid restrictions on the given initial data. These compatibilityconditions have no physical meaning and, therefore, they can beavoided, if the solution is sought in larger spaces, for instance inweighted H\"older spaces. We give general existence and uniqueness result andprovide some examples of applications of the main theorem. The maintool is the monotone iterations method. Preliminary we developed thelinear theory with existence and comparison results. The principleuse of the positivity lemma is the construction of a monotonesequences for our problem. Initial iteration may be taken as eitheran upper solution or a lower solution. We provide some examples ofupper and lower solution for the case of linear equations andquasilinear boundary conditions. We notice that this approach canalso be extended to other problems and systems of fractionalequations as soon as we will be able to construct appropriate upperand lower solutions.