We study the initial value problem of the exponential wave equation in $math{R}^{n+1}$ for small initial data. We shows, in the case of $n=1$, the global existence of solution by applying the formulation of first order quasilinear hyperbolic system which is weakly linearly degenerate. When $n{geq}2$, a vector field method is applied to show the stability of a trivial solution ${phi}=0$.