Electronic Journal of Differential Equations,
Vol. 1995(1995), No. 09, pp. 1--15.
Title: Singularity Formation in Systems of Non-strictly
Hyperbolic Equations
Authors: R. Saxton (Univ. of New Orleans, LA, USA)
V. Vinod (Univ. of New Orleans, LA, USA)
Abstract:We analyze finite time singularity formation for two systems
of hyperbolic equations. Our results extend previous proofs of breakdown
concerning $2\times 2$ non-strictly hyperbolic systems to $n \times n$
systems, and to a situation where, additionally, the condition of
genuine nonlinearity is violated throughout phase space. The systems
we consider include as special cases those examined by Keyfitz and
Kranzer and by Serre. They take the form
$$ u_{t} + (\phi(u)u)_{x} = 0, $$
where $\phi$ is a scalar-valued function of the $n$-dimensional
vector $u$, and
$$ u_{t}+\Lambda(u)u_{x} = 0, $$
under the assumption $\Lambda = {\rm diag}\,
\{\lambda^{1},\ldots,\lambda^{n}\}$ with
$\lambda^{i}=\lambda^{i}(u-u^{i})$,
where
$u-u^{i}\equiv\{u^{1},\ldots,u^{i-1},u^{i+1},\ldots,u^{n}\}$.
Submitted June 12, 1995. Published June 28, 1995.
Math Subject Classification: 35L45, 35L65, 35L67, 35L80.
Key Words: Finite time breakdown; non-strict hyperbolicity;
linear degeneracy.