The density of the $(\alpha,\beta)$-superprocess and singular solutions to a fractional non-linear PDE

We consider the density $X_t(x)$ of the critical $(\alpha,\beta)$-superprocess in $R^d$ with $\alpha\in (0,2)$ and $\beta<\frac \alpha d$. A recent result from PDE implies a dichotomy for the density: for fixed $x$, $X_t(x)>0$ a.s. on $\{X_t\neq 0\}$ if and only if $\beta \leq \beta^*(\alpha) = \frac{\alpha}{d+\alpha}$. We strengthen this and show that in the continuous density regime, $\beta < \beta^*(\alpha)$ implies that the density function is strictly positive a.s. on $\{X_t\neq 0\}$. We then give close to sharp conditions on a measure $\mu$ such that $\mu(X_t):=\int X_t(x)\mu(dx)>0$ a.s. on $\{X_t\neq 0 \}$. Our characterization is based on the size of $supp(\mu)$, in the sense of Hausdorff measure and dimension. For $s \in [0,d]$, if $\beta \leq \beta^*(\alpha,s)=\frac{\alpha}{d-s+\alpha}$ and $supp(\mu)$ has positive $x^s$-Hausdorff measure, then $\mu(X_t)>0$ a.s. on $\{X_t\neq 0\}$; and when $\beta > \beta^*(\alpha,s)$, if $\mu$ satisfies a uniform lower density condition which implies $dim(supp(\mu)) < s$, then $P(\mu(X_t)=0|X_t\neq 0)>0$. We also give new result for the fractional PDE $\partial_t u(t,x) = -(-\Delta)^{\alpha/2}u(t,x)-u(t,x)^{1+\beta}$ with domain $(t,x)\in (0,\infty)\times R^d$. The initial trace of a solution $u_t(x)$ is a pair $(S,\nu)$, where the singular set $S$ is a closed set around which local integrals of $u_t(x)$ diverge as $t \to 0$, and $\nu$ is a Radon measure which gives the limiting behaviour of $u_t(x)$ on $S^c$ as $t\to 0$. We characterize the existence of solutions with initial trace $(S,0)$ in terms of a parameter called the saturation dimension, $d_{sat}=d+\alpha(1-\beta^{-1})$. For $S\neq R^d$ with $dim(S)> d_{sat}$ (and in some cases $dim(S)=d_{sat}$) we prove that no such solution exists. When $dim(S)<d_{sat}$ and $S$ is the compact support of a measure satisfying a uniform lower density condition, we prove that a solution exists.