ASYMPTOTIC SHAPES OF SHOCK CURVES FOR SOME CRITICAL CASES

Abstract

We consider the scalar conservation law with convex flux. When the antiderivative Φ of the initial data satisfies Φ(∞) > inf y∈ℝΦ(y), Lax obtained that asymptotic shapes of shock curves at the far ends are [Formula: see text] with some constant C. In this paper, we consider the critical case when Φ(∞) = inf y∈ℝΦ(y), and show that asymptotic shapes of shock curves are [Formula: see text] instead of [Formula: see text]. We give the formula for asymptotic curves, and provide examples showing rich phenomena of the critical case.