Tsukuba Journal of Mathematics

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

Tamotu Kinoshita

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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


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