| Abstract: |
| In this paper, we prove the existence of an initial trace ${\mathcal T}_u$ for any positive solution $u$ to the semilinear fractional diffusion equation $(H)$
$$\partial_t u + (-\Delta)^\alpha u+f(t,x,u)=0\quad {\rm in}\quad (0,+\infty)\times\mathbb R^N,$$
where $N\geq1$, the operator $(-\Delta)^\alpha$ with $\alpha\in(0,1)$ is the fractional Laplacian, $f:\mathbb R_+\times\mathbb R^N\times\mathbb R_+\rightarrow\mathbb R$ is a Caratheodory function satisfying $f(t,x,u)u\geq 0$ for all $(t,x,u)\in \mathbb R_+\times\mathbb R^N\times\mathbb R_+$ and $\mathbb R_+=[0,+\infty)$.
We define the regular set of the trace ${\mathcal T}_u$ as an open subset ${\mathcal R}_u\subset\mathbb R^N$ carrying a nonnegative Radon measure $\nu_u$ such that
$$\lim_{t\to 0}\int_{{\mathcal R}_u}u(t,x)\zeta(x)\, dx=\int_{{\mathcal R}_u}\zeta\, d\nu_u,\quad\forall\, \zeta\in C^{2}_0({\mathcal R}_u),
$$
and the singular set ${\mathcal S}_u=\R^N\setminus {\mathcal R}_u$ as the set points $a$ such that
$$\limsup_{t\to 0}\int_{B_\gr(a)}u(t,x)\, dx=+\infty\quad{\rm for\ any}\ \, \gr>0.
$$
We also study the reverse problem of constructing a positive solution to $(H)$ with a given initial trace $(\mathcal S,\nu)$, where $\mathcal S\subset\mathbb R^N$ is a closed set and $\nu$ is a positive Radon measure on $\mathcal R=\mathbb R^N\setminus\mathcal S$ and develop the case $f(t,x,u)=t^\beta u^p$ with $\beta>-1$ and
$p>1$. |
|