Asymptotic behaviors of the solution to an initial-boundary value problem for scalar viscous conservation laws

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Abstract

This paper is concerned with the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws ut + f(u)x = uxx on [0, 1], with the boundary condition u(0, t) = u(t) → u, u(1, t) = u+(t) → u+, as t → +∞ and the initial data u(x,0) = u0(x) satisfying u0(0) = u(0), u0(1) = u+(1), where u± are given constants, uu+ and f is a given function satisfying f″(u) > 0 for u under consideration. By means of an elementary energy estimates method, both the global existence and the asymptotic behavior are obtained. When uu+, which corresponds to rarefaction waves in inviscid conservation laws, no smallness conditions are needed. While for u > u+, which corresponds to shock waves in inviscid conservation laws, it is established for weak shock waves, that is, |uu+| is small. Moreover, when u±(t) ≡ u±, t ≥ 0, exponential decay rates are both obtained.

Keywords

Viscous conservation laws
Asymptotic behavior
Finite interval

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The work for the first author is supported by the National Natural Science Foundation of China 10061001 and Guangxi Natural Science Foundation 9912020.

The first author is grateful to K. Nishihara of Waseda University and Dr. J. Quansen of Capital Normal University for their helpful discussion and the warm encouragement.

The work for the second author is supported by the National Natural Science Foundation of China 19901012.

The second author is grateful to J. Wang and L. Hsiao for their helpful discussion and the warm encouragement when she visited the Academy of Mathematics and System Science, Chinese Academy of Science.