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We consider non-negative solutions to the semilinear space-fractionaldiffusion problem $(\partial_t+(-\Delta)^{\alpha/2})u=\rho(x)u^p$ on wholespace $R^n$ with nonnegative initial data and with $(-\Delta)^{\alpha/2}$ beingthe $\alpha$-Laplacian operator, $\alpha\in (0,2)$. Here $p>0$ and $\rho(x)$ isa non-negative locally integrable function. For $\rho(x)=1$ we show that thefujita exponent is $p_F=1+\frac{\alpha}{n}$ and the Liouville type result forthe 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 onepositive solution. This existence result is proved after we establish auniqueness result about solutions of fractional Poisson equation.

Liouville theorems and Fujita exponent for nonlinear space fractional diffusions