Logarithmic convexity of non-symmetric time-fractional diffusion equations

TL;DR

This paper extends the concept of logarithmic convexity to non-symmetric time-fractional diffusion equations with Caputo derivatives, under specific conditions, and validates the results through numerical experiments.

Contribution

It demonstrates that logarithmic convexity applies to non-symmetric fractional diffusion equations when the drift is a gradient field, expanding previous symmetric case results.

Findings

Logarithmic convexity holds for non-symmetric fractional diffusion with gradient drift.

Numerical experiments confirm theoretical predictions.

The study opens avenues for further research on non-symmetric fractional PDEs.

Abstract

We consider a class of diffusion equations with the Caputo time-fractional derivative $\partial_t^\alpha u=L u$ subject to the homogeneous Dirichlet boundary conditions. Here, we consider a fractional order $0<\alpha < 1$ and a second-order operator $L$ which is elliptic and non-symmetric. In this paper, we show that the logarithmic convexity extends to this non-symmetric case provided that the drift coefficient is given by a gradient vector field. Next, we perform some numerical experiments to validate the theoretical results in both symmetric and non-symmetric cases. Finally, some conclusions and open problems will be mentioned.