Asymptotics for a wave equation with critical exponential nonlinearity

Abstract

In this paper, we discuss some qualitative analysis of solutions to the following Cauchy problem of wave equations involving the 1/2-Laplace operator with critical exponential nonlinearity [Formula presented] where λ>0, δ≥0, q>2, and α0>0. By using the contraction mapping principle, we show that the above Cauchy problem has a unique local solution. With the help of the potential well argument, we characterize the stable sets by the asymptotic behavior of solutions as t goes to infinity, as well as the unstable sets by the blow-up of solutions in finite time.