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The main objective of this paper is analysis of the initial-boundary valueproblems for the linear time-fractional diffusion equations with a uniformlyelliptic spatial differential operator of the second order and the Caputo typetime-fractional derivative acting in the fractional Sobolev spaces. Theboundary conditions are formulated in form of the homogeneous Neumann or Robinconditions. First we discuss the uniqueness and existence of solutions to theseinitial-boundary value problems. Under some suitable conditions on the problemdata, we then prove positivity of the solutions. Based on these results,several comparison principles for the solutions to the initial-boundary valueproblems for the linear time-fractional diffusion equations are derived.

Comparison principles for the time-fractional diffusion equations with the Robin boundary conditions. Part I: Linear equations