## Advances in Differential Equations

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

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

#### 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