Abstract
Let v=v(x) be a non-trivial bounded steady solution of a viscous scalar conservation law u t+f(u) x =u xx on a half-line R+, with a Dirichlet boundary condition. The semi-group of this IBVP is known to be contractive for the distance d(u, u′)≔‖u−u′‖1 induced by L 1(R+). We prove here that v is asymptotically stable with respect to d: if u 0−v∈L 1, then ‖u(t)−v‖1→0 as t→+∞. When v is a constant, we show that this property holds if and only if f′(v)≤0. These results complement our study of the Cauchy problem [2].
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Freistühler, H., Serre, D. The L1-Stability of Boundary Layers for Scalar Viscous Conservation Laws. Journal of Dynamics and Differential Equations 13, 745–755 (2001). https://doi.org/10.1023/A:1016646026758
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DOI: https://doi.org/10.1023/A:1016646026758