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

Abstract

Consider the second order nonlinear partial differential equation: $ \partial_t^2 u = F(u, \partial_x u), \quad (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 $ \varepsilon $. 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.

Keywords

• cauchy problem
• lifespan of solutions
• gevrey functions
• small data