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Sliding methods for tempered fractional parabolic problem
- Part of
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- Journal:
- Canadian Journal of Mathematics / Volume 76 / Issue 4 / August 2024
- Published online by Cambridge University Press:
- 28 July 2023, pp. 1358-1378
- Print publication:
- August 2024
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- Article
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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.
