Boundary value problems for fractional diffusion equations

Over the past few decades an intensive effort has been put into developing theoretical models for systems with diffusive motion that can not be modelled as standard Brownian motion. The sig-nature of this anomalous diffusion is that the mean square displacement of the diffusing species $\langle(\Delta \mathbf{x})^{2}\rangle$ scales as a nonlinear power law in time, i.e. $\langle(\Delta \mathbf{x})^{2}\rangle\sim t^{\alpha}$. If $\alpha\in(0,1)$, this is referred to as subdiffusion. In recent years the additional motivation for these studies has been stimulated by experimental measurements of subdiffusion in porous media, glass forming materials, biological media. The review paper by Klafter et al. provides numerous references to physical phenomena in which anomalous diffusion occurs.

We analyze classical solvability of some initial-boundary value problems for the subdiffusion linear and nonlinear equations with different boundary conditions.