Global Existence and Blow-up of Solutions for a System of Fractional Wave Equations

Abstract

We investigate the Cauchy problem for a $2 \times 2$-system of weakly coupled semi-linear fractional wave equations with polynomial nonlinearities posed in $\mathbb{R}^{+} \times \mathbb{R}^{N}$. Under appropriate conditions on the exponents and the fractional orders of the time derivatives, it is shown that there exists a threshold value of the dimension $N$, for which, small data-global solutions as well as finite time blowing-up solutions exist. Furthermore, we investigate the $L^{\infty}$-decay estimates of global solutions.