AIMS Mathematics, 2018, 3(1): 35-43. doi: 10.3934/Math.2018.1.35

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Asymptotic stability of degenerate stationary solution to a system of viscous conservation laws in half line

Department of Applied Mathematics, Kumamoto University, Kurokami 2-39-1, Chuo-ku, Kumamoto 860-8555, Japan

In this paper, we study a system of viscous conservation laws given by a form of a symmetric parabolic system. We consider the system in the one-dimensional half space and show existence of a degenerate stationary solution which exists in the case that one characteristic speed is equal to zero. Then we show the uniform a priori estimate of the perturbation which gives the asymptotic stability of the degenerate stationary solution. The main aim of the present paper is to show the a priori estimate without assuming the negativity of non-zero characteristics. The key to proof is to utilize the Hardy inequality in the estimate of low order terms.
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© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0

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