Monotone iterations method for fractional diffusion equations

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