We prove the uniqueness of the positive solutions of the following elliptic system: (1) $-\Delta(u(x))=u(x)^{\alpha}v(x)^{\beta}$, (2) $-\Delta(v(x))=u(x)^{\beta}v(x)^{\alpha}$. Here $x\in R^n$, $n\g...
We prove that the nonexistence of supersonic finite-energy traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity is a general phenomenon which holds for a large...
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Stability of Traveling Waves in Quasi-Linear Hyperbolic Systems with Relaxation and Diffusion
SIAM J. Math. Anal. Volume 40, Issue 3, pp. 1058-1075 (2008)
Published October 17, 2008
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We establish the existence and the stability of traveling wave solutions of a quasi-linear hyperbolic system with both relaxation and diffusion. The traveling wave solutions are shown to be asymptotically stable under small disturbances and under the subcharacteristic condition using a weighted energy method. The delicate balance between the relaxation and the diffusion that leads to the stability of the traveling waves is identified; namely, the diffusion coefficient is bounded by a constant multiple of the relaxation time. Such a result provides an important first step toward the understanding of the transition from stability to instability as parameters vary in physical problems involving both relaxation and diffusion.
©2008 Society for Industrial and Applied Mathematics| History: | Received May 6, 2007; accepted June 20, 2008; published October 17, 2008 |
| Permalink: | http://dx.doi.org/10.1137/070690638 |
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SIAM