On the large-time asymptotic behaviors of λ-dissipative solutions to the Hunter–Saxton equation

This paper studies the large-time asymptotic behavior of the λ-dissipative solutions (λ∈[0,1]) to the Hunter–Saxton equation by using the explicit characteristics. The leading order term in the large-time asymptotic expansions in both spaces L∞(R) and H˙1(R) is shown to be given by a special self-similar solution usually referred to as the kink wave, which is determined by the total remaining energy after accounting for all possible sudden energy releases (blow-ups) in the system. This remaining energy can be calculated based on the initial total energy, the singular measures resulting from the initial data, and the dissipation parameter λ. Importantly, our results encompass both the energy conservative (λ=0) and energy dissipative (λ=1) solutions.