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2005 The Cauchy problem for quasilinear hyperbolic equations with non-absolutely continuous coefficients in the time variable
Alessia Ascanelli
Adv. Differential Equations 10(10): 1165-1181 (2005).

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.

Citation

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Alessia Ascanelli. "The Cauchy problem for quasilinear hyperbolic equations with non-absolutely continuous coefficients in the time variable." Adv. Differential Equations 10 (10) 1165 - 1181, 2005.

Information

Published: 2005
First available in Project Euclid: 18 December 2012

zbMATH: 1122.35083
MathSciNet: MR2162365

Subjects:
Primary: 35L80
Secondary: 35L75

Rights: Copyright © 2005 Khayyam Publishing, Inc.

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Vol.10 • No. 10 • 2005
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