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

Abstract

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 $F(u)=B_1(u,...,u),G(v)=B_2(v,...,v)$