Global Existence for a Nonlinear System with Fractional Laplacian in Banach Space

We consider the cauchy problem for the fractional power dissipative equation $u_t+(-\Delta )^{\beta/2} u=F(u)$, where $\beta>0$ and $F(u)=B(u, ...,u)$ and $B$ is a multilinear form on a Banach space $E$. We show a global existence result assuming some properties of scaling degree of the multilinear form and the norm of the space $E$. We extend the ideas used for the treating of the equation to determine the global existence for the system $u_t+(-\Delta)^{\beta/2}= F(v)$, $v_t+(-\Delta )^{\beta/2}v=G(u)$ where $F(u)=B_1(u,...,u), G(v)=B_2(v,...,v)$