Advances in Differential Equations

The Cauchy problem for quasilinear hyperbolic equations with non-absolutely continuous coefficients in the time variable

Alessia Ascanelli

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We consider the Cauchy problem \[\left\{ \begin{array}{lll} P\bigl(t,x,D^{m-1}u,D_{t},D_{x}\bigr)u(t,x)=f(t,x,D^{m-1}u) \\ \partial^j_{t} u(0,x)=u_{j}(x),\ \ \ j=0,...,m-1, \end{array} \right.\] in $[0,T]\times{\mathbb{R}}^n$ for a quasilinear weakly hyperbolic operator \[ P\bigl(t,x,D^{m-1}u,D_{t},D_{x}\bigr)=D_{t}^{m}+\sum_{j=0}^{m-1} \sum_{|\alpha|={m-j}}a_{\alpha}^{(j)}(t,x,D^{m-1}u)D_x^{\alpha}D_t^j\] with coefficients $a_{\alpha}^{(j)}$ having the first time derivative with singular behavior of the type $t^{-q}$, $q>1$, as $t\to 0$. We show that for $t\leq T_0^\ast$, $T_0^\ast$ sufficiently small, given Cauchy data in a Gevrey class $G^\sigma$ there exists a unique solution $u\in C^{m-1}([0,T_0^\ast];G^\sigma)$ provided that $\sigma < \frac{qr}{qr-1}$ where $r$ denotes the largest multiplicity of the characteristic roots.

Article information

Source
Adv. Differential Equations Volume 10, Number 10 (2005), 1165-1181.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867808

Mathematical Reviews number (MathSciNet)
MR2162365

Zentralblatt MATH identifier
1122.35083

Subjects
Primary: 35L80: Degenerate hyperbolic equations
Secondary: 35L75: Nonlinear higher-order hyperbolic equations

Citation

Ascanelli, Alessia. The Cauchy problem for quasilinear hyperbolic equations with non-absolutely continuous coefficients in the time variable. Adv. Differential Equations 10 (2005), no. 10, 1165--1181. https://projecteuclid.org/euclid.ade/1355867808.


Export citation