On the Cauchy problem for a weakly dissipative μ-Degasperis-Procesi equation

Abstract

In this paper, we study the Cauchy problem of a weakly dissipative

Global Weak Solutions for the Weakly Dissipative μ-Hunter–Saxton Equation

“…Firstly, Lemma 2.1 and u = g * y, g ≥ 0 imply y(t, x) and u(t, x) have the same sign with y 0 (x). Moreover, from the proof of Theorem 5.1 in [10], we have u x (t, x) ≥ −|µ 0 |. Now note that given t ∈ [0, T ), there is a ξ(t) ∈ S such that u x (t, ξ(t)) = 0 by the periodicity of u to x-variable.…”

“…Proof. Except for (4) and 5, all of the conclusions in Lemma 2.2 can be found in [10]. So we only need to prove (4) and (5) here.…”

“…In particular, µ(g) = 1. [10] ensures the existence of a maximal T > 0 and a solution u to (1.2) such that…”

“…Here the constant λ is assumed to be positive and λy = λ(µ(u) − u xx ) is the weakly dissipative term. The Cauchy problem (1.1) has been discussed in [10] recently. The author established the local well-posedness, derived the precise blow-up scenario for Eq.…”