## Tsukuba Journal of Mathematics

- Tsukuba J. Math.
- Volume 21, Number 2 (1997), 397-420.

### 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.

#### Article information

**Source**

Tsukuba J. Math., Volume 21, Number 2 (1997), 397-420.

**Dates**

First available in Project Euclid: 30 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1496163249

**Digital Object Identifier**

doi:10.21099/tkbjm/1496163249

**Mathematical Reviews number (MathSciNet)**

MR1473930

**Zentralblatt MATH identifier**

0897.35047

#### Citation

Kinoshita, Tamotu. On the Cauchy Problem with small analytic data for nonlinear weakly hyperbolic systems. Tsukuba J. Math. 21 (1997), no. 2, 397--420. doi:10.21099/tkbjm/1496163249. https://projecteuclid.org/euclid.tkbjm/1496163249