Instability of Discontinuous Traveling Waves for Hyperbolic Balance Laws

✍ Scribed by Carlo Sinestrari

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 356 KB
• Volume: 134
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.3223

✦ Synopsis

It is known that scalar hyperbolic conservation laws with source term and periodic initial value have a property of Poincare Bendixson type, namely the solutions converge either to a constant state or to a periodic traveling wave, which is necessarily discontinuous. In this paper we show that generically (with respect to the L 1 topology) the solutions exhibit a behaviour of the former type. We also show that, while the rate of convergence to a constant state is exponential, the convergence to a traveling wave can be arbitrarily slow.

1997 Academic Press Fan and Hale [4] have also shown that the global attractor of the equation consists of orbits connecting either two constant states, or two traveling waves, or a constant state to a traveling wave or vice versa.