Liouville theorems and Fujita exponent for nonlinear space fractional diffusions

We consider non-negative solutions to the semilinear space-fractional diffusion problem $(\partial_t+(-\Delta)^{\alpha/2})u=\rho(x)u^p$ on whole space $R^n$ with nonnegative initial data and with $(-\Delta)^{\alpha/2}$ being the $\alpha$-Laplacian operator, $\alpha\in (0,2)$. Here $p>0$ and $\rho(x)$ is a non-negative locally integrable function. For $\rho(x)=1$ we show that the fujita exponent is $p_F=1+\frac{\alpha}{n}$ and the Liouville type result for the stationary equation is true for $0<p\leq 1+\frac{\alpha}{n-\alpha}$. When $p=1/2$ and $\rho(x)$ satisfies an integrable condition, there is at least one positive solution. This existence result is proved after we establish a uniqueness result about solutions of fractional Poisson equation.