Abstract
This paper is concerned with the long time behaviour of a weaklydissipative Degasperis-Procesi equation. Our analysis discloses theco-existence of global in time solutions and finite time break downof strong solutions. Our blow-up criterion for the initial profilegeneralizes considerably results obtained earlier in [32].
Highlights
IntroductionWe our analysis shows that the blow-up rate of the Degasperis-Procesi equation is not affected by the weakly dissipative term, but the occurrence of blowup of Eq(1.2) is affected by the dissipative parameter
The Degasperis-Procesi equation ut − utxx + 4uux = 3uxuxx + uuxxx, t > 0, x ∈ R (1.1)arises in the shallow-water medium-amplitude regime [1, 13, 22], introduced to capture stronger nonlinear effects that will allow for breaking waves, since the latter are not modeled by the shallow-water small-amplitude regime characteristic for the KdV equation
We found that concerning local well-posedness and blow-up phenomena the behaviour of Eq(1.2) is comparable to that of the Degasperis-Procesi equation
Summary
We our analysis shows that the blow-up rate of the Degasperis-Procesi equation is not affected by the weakly dissipative term, but the occurrence of blowup of Eq(1.2) is affected by the dissipative parameter Another difference between Eq(1.2) and Eq(1.1) is the fact that the dissipation annihilates the conservation laws: E1(u) = ydx, E2(u) = yvdx, E3(u) = u3dx, R where y = (1 − ∂x2)u and v = (4 − ∂x2)−1u, which play an important role in the study of Eq(1.1). We recall the local well-posedness result, the precise blow-up scenario of the weakly dissipative Degasperis-Procesi equation (1.2) and several useful lemmas from [32] which will be needed in the sequel. We establish a criterion, guaranteeing global existence theorem and a new blow-up result for strong solutions to Eq(2.1) with certain profiles. We will determine the blow-up rate and the blow-up set for solutions which do not exist globally in time
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
