Abstract
In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation. The local well-posedness of Cauchy problem is investigated by utilizing Kato’s theorem. The blow-up criteria and the blow-up rate are established by applying monotonicity. Finally, the global existence results for solutions to the Cauchy problem of equation are proved by structuring functions.
Highlights
In this paper, we consider the Cauchy problem of periodic two-component Camassa-Holm equation with a generalized weakly dissipation:= = uuρtt(−0+,u(xxρ)xtu+)uxk= 0u(x x0+,) ;3ρuu( x0,−x2)u x uxx ρ0 − uuxxx (x), + λ (u − uxx ) σρρ x =u (t, x)= u (t, x +1); ρ (t, x)= ρ (t, x +1), t > 0, x ∈ R, t > 0, x ∈ R,(1.1) x ∈ R, t ≥ 0, x ∈ R, where λ ≥ 0 and k is a fixed constant; σ is a free parameter
In this paper we study a periodic two-component Camassa-Holm equation with generalized weakly dissipation
The blow-up criteria and the blow-up rate are established by applying monotonicity
Summary
We consider the Cauchy problem of periodic two-component Camassa-Holm equation with a generalized weakly dissipation:. Constantin and Ivanov [6] investigated the global existence and blow-up phenomena of strong solutions of Equation (1.2). It is necessary to study periodic two-Camassa-Holm equation with a generalized weakly dissipation. Hu and Yin [11] study the blow-up of solutions to a weakly dissipative periodic rod equation. Hu considered global existence and blow-up phenomena for a weakly dissipative two-component Camassa-Holm system [12] [13]. The results show that the behavior of solutions to the periodic two-component Camassa-Holm equation with a generalized weakly dissipation is similar to Equation (1.2) and the blow-up rate of Equation (1.1) is not affected by the dissipative term when σ > 0.
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