Global existence, blow-up in finite time and vacuum isolating phenomena for a system of semilinear wave equations associated with the helical flows of Maxwell fluid

Abstract

In this paper, we investigate the initial boundary value problem for the system of semilinear wave equations associated with the helical flows of Maxwell fluid. We introduce a family of potential wells and discuss the invariant sets and vacuum isolating behavior of solutions. Using the potential well argument, we show the global existence, finite time blow-up, the asymptotic behavior of solutions, estimate the lifespan as well as blow-up rate of the weak solution. In particular, we establish a sharp criterion for global existence and blow-up of solutions when E0⩽d. Finally, when the initial energy is supercritical, we give some explicit criterion for blow-up in finite.