Local and nonlocal boundary value problems for generalized Aller–Lykov equation

In mathematical modelling of solid media with memory there arise equations describing a new type of wave motion, which is between the usual diffusion and classical waves. Here we mean differential equations with fractional derivatives both in time and spatial variables, which are a base for most part of mathematical models in mechanics of liquids, viscoelasticity as well as in processes in media with fractal structure and memory.
In the present work we present a qualitatively new moisture transfer equation being a generalization of Aller-Lykov equation. This generalization provides an opportunity to reflect specific features of the studied objects in the nature of the equation such, namely, the structure and physical properties of the going processes, by means of introducing a fractal velocity of moisture varying.
The work is devoted to studying local and nonlocal boundary value problems for inhomogeneous Aller-Lykov moisture transfer equation with variable coefficients and Riemann-Liouville fractional time derivative. For a generalized equation of Aller-Lykov type we consider initial boundary value problems with Dirichlet and Robin boundary conditions as well as nonlocal problems involving nonlocality in time in the boundary conditions. Assuming the existence of regular solutions, by the method of energy inequalities, we obtain apriori estimates in terms of Riemann-Liouville fractional derivative, which imply the uniqueness of the solutions to the considered problems as well as their stability in the right hand side and initial data.