On the Cauchy Problem with small analytic data for nonlinear weakly hyperbolic systems

Abstract

In this paper we investigate the life span of the Cauchy Problem for nonlinear systems of the form $(*)$ $\left\{\begin{array}{l}\partial_{t}u=f(t,x,u,\partial_{1}u,\ldots,\partial_{n}u)\\u(0,x)=\epsilon\phi(x).\end{array}\right.$ Assuming that $(*)$ is weakly hyperbolic and has the solution $u\equiv 0$ with $\phi\equiv 0$, we prove that i) lifespan $T_{\epsilon}\rightarrow\in\infty$ as $\epsilon\rightarrow 0$. $T_{\epsilon}$ admits the asymptotic estimate $T_{\epsilon}\geq\psi^{-1}(\mu\log\log(1/\epsilon)$, where $\psi(t)=\int_{0^{t}}|f(\tau)|d\tau,\mu \gt 0$. ii) $u=0$ is a stable solution. In order to get this fact, we first consider the case of linear systems and then apply to nonlinear systems.