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Consider the Dirichlet problem associated to the classical heat equation $\partial_t u - \Delta u = 0$ and to the fractional heat equation $\partial_t u + (- \Delta)^s u = 0$.

  • Where can I find a reference on semigroup theoretic approach to these problems?

In particular, where can I find a proof that there exists one and only one solution to

$$\partial_t u + (- \Delta)^s u = 0 \;\text{ in } \Omega,\quad t>0$$ $$u=0 \;\text{ in } \mathbb{R}^n \setminus \Omega, \quad t>0$$ $$u(t,x) = u_0(x) \;\text{ in } \Omega, \quad t=0,$$

which is given by $$u(t,x) = (T(t)u_0)(x), \overline{\Omega}, t \ge 0$$ where $(T)_{t \ge 0}$ is a strongly continuous semigroup (on $C$, $BUC$, or $C_b$ or one $L^p$ space) and the solution continuously depends on the initial datum?

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  • $\begingroup$ For the Laplacian, this is very standard. Recently I wrote a paper about various definitions of the fractional Laplace operator (arXiv), with a section on semigroup approach; it does not contain any proof, but it points to some references that might be relevant. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 19, 2017 at 19:11
  • $\begingroup$ @MateuszKwaśnicki Thank you for your comment. I edited my question to make it more specific. $\endgroup$
    – user103450
    Commented Oct 19, 2017 at 19:22
  • $\begingroup$ Oh, so you mean the Dirichlet problem. Now you need some regularity of $\Omega$ for the existence of the solution, and a maximal principle for uniqueness. You also need to specify in what sense the fractional Laplacian is understood, and in what function space your solutions are. I guess you will find a lot of information here and here. For a more general approach, you can consult potential-theoretic literature (e.g. Bliedtner–Hansen book), but this will be a hard read. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 19, 2017 at 20:22
  • $\begingroup$ Try "Long-term behavior of reaction-diffusion equations with nonlocal boundary conditions on rough domains" located here among other places arxiv.org/abs/1503.05744 $\endgroup$
    – SetHead
    Commented Oct 19, 2017 at 21:56
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    $\begingroup$ Shouldn't it be $(-\Delta)^s$ instead of $-\Delta^s$ ? $\endgroup$
    – Johannes Hahn
    Commented Oct 29, 2017 at 17:43

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