Lifespan of solutions to second order Cauchy problems with small Gevrey data

Consider the second order nonlinear partial differential equation:

$${\mathrm{\partial }}_t^{2}u=F(u,{\mathrm{\partial }}_{x}u),(t,x)\in \mathbb{C}\times \mathbb{R}.$$

Given small analytic data, Yamane was able to obtain the order of the lifespan of the solution with respect to the smallness parameter $\epsilon$. On the other hand, Gourdin and Mechab studied the lifespan of the solution given small Gevrey data, but under the assumption that $F$ is independent of $u$. In this paper, we considered non-vanishing Gevrey data and used the method of successive approximations to obtain a solution and constructively estimate its lifespan.