Diffusion equations with general nonlocal time and space derivatives

Abstract

In the present study, firstly, based on the continuous time random walk (CTRW) theory, general diffusion equations are derived. The time derivative is taken as the general Caputo-type derivative introduced by Kochubei and the spatial derivative is the general Laplacian defined by removing the conditions (1.5) and (1.6) from the definition of the general fractional Laplacian proposed in the paper (Servadei and Valdinoci, 2012). Secondly, the existence of solutions of the Cauchy problem for the general diffusion equation is proved by extending the domain of the general Laplacian to a general Sobolev space. The results for positivity and boundedness of the solutions are also obtained. In the last, the existence result for solutions of the initial boundary value problem (IBVP) for the general diffusion equation on a bounded domain is established by using the Friedrichs extension of the general fractional Laplacian introduced in the book (Bisci, Radulescu and Servadei, 2016).

References in corpus (7)

• Anomalous transport in the crowded world of biological cells
• The fundamental solution of the space-time fractional diffusion equation
• Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients
• Weak ergodicity breaking of receptor motion in living cells stemming from random diffusivity
• Aging dynamics in interacting many-body systems
• Fractional Brownian motion with a reflecting wall
• Fraction diffusion: Recovering the distributed fractional derivative from overposed data

Cited by in corpus (2)

• Cauchy problem for general time fractional diffusion equation
• Global well-posedness of space-time fractional diffusion equation with Rockland operator on graded Lie group