Liouville theorems and Fujita exponent for nonlinear space fractional diffusions

Abstract

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 $0<p\le 1+\frac{\alpha }{n-\alpha }$. 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.

16 pages

References in corpus (3)

• The fundamental solution of the space-time fractional diffusion equation
• Sharp criteria of Liouville type for some nonlinear systems
• Fractional Diffusion Processes: Probability Distributions and Continuous Time Random Walk