Abstract
In this paper, we study the Cauchy problem of a two-component b-family system which arises in shallow water theory. We first derive the precise blow-up scenario and present a blow-up result. Then we investigate the infinite propagation speed in the sense that the corresponding solution with compact supported initial datum does not have compact spatial support any longer in its lifespan.MSC: 35G25, 35L05, 35Q58.
Highlights
In this paper, we consider the following nonlinear system: ⎧ ⎪⎪⎪⎨ ut ρt – +utxx + (b +x =,)uux buxuxx uuxxx + σρρx =⎪⎪⎪⎩ u(, x) = u (x), ρ(, x) = ρ (x), t >, x ∈ R, t >, x ∈ R, x ∈ R, x ∈ R
When b =, we find the Degasperis-Procesi equation [ ] from ( . ), which is regarded as a model for nonlinear shallow water dynamics
The aim of this paper is to present a blow-up result of solutions to ( . ) with the case of σ = and to examine the propagation behavior of compactly supported solutions to ( . ) with σ =, namely whether solutions which are initially compactly supported will retain this property throughout their time of evolution
Summary
) reduces to the following b-family equation, which was extensively studied in [ – ]: mt + umx + buxm = , t > , x ∈ R, ) becomes the Camassa-Holm equation, modeling the unidirectional propagation of shallow water waves over a flat bottom. ) on the line and on the circle, and established the local well-posedness, described the precise blow-up scenario, proved that the equation has strong solutions which exist globally in time and blow up in finite time.
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