This paper is concerned with the asymptotic behaviors of the solutions to the initial-boundary value problem for scalar viscous conservations laws u(t) + f(u)(x) = u(xx) on [0,1], with the boundary condition u(0, t) = u(-)(t) --> u(-), u(1, t) = u(+)(t) --> u(+), as t --> +infinity and the initial data u(x,0) = u(0)(x) satisfying u(0)(0) = u(-)(0), u(0)(1) = u(+)(1), where u(+/-) are given constants, u(-) not equal u(+) and f is a given function satisfying f"(u) > 0 for u tinder consideration. By means of an elementary energy estimates method, both the global existence and the asymptotic behavior are obtained. When u(-) < u(+), 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, u(-) - u(+) is small. Moreover, when u(+/-)(t) equivalent to u(+/-),t greater than or equal to 0, exponential decay rates are both obtained. (C) 2002 Elsevier Science Ltd. All rights reserved. [References: 9]
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