Sliding methods for tempered fractional parabolic problem

Peng, Shaolong

Canadian journal of mathematics, Tome 76 (2024) no. 4, pp. 1358-1378

Résumé

In this article, we are concerned with the tempered fractional parabolic problem $$ \begin{align*}\frac{\partial u}{\partial t}(x, t)-\left(\Delta+\lambda\right)^{\frac{\alpha}{2}} u(x, t)=f(u(x, t)), \end{align*} $$where $-\left (\Delta +\lambda \right )^{\frac {\alpha }{2}}$ is a tempered fractional operator with $\alpha \in (0,2)$ and $\lambda $ is a sufficiently small positive constant. We first establish maximum principle principles for problems involving tempered fractional parabolic operators. And then, we develop the direct sliding methods for the tempered fractional parabolic problem, and discuss how they can be used to establish monotonicity results of solutions to the tempered fractional parabolic problem in various domains. We believe that our theory and methods can be conveniently applied to study parabolic problems involving other nonlocal operators.

Détail

DOI : 10.4153/S0008414X23000457

Mots-clés : Tempered fractional parabolic problem, maximum principle, sliding methods, monotonicity