Blow-up to a shallow water wave model including the Degasperis-Procesi equation

Jin Hong; Shaoyong Lai; Jin Hong; Shaoyong Lai

doi:10.3934/math.20231296

AIMS Mathematics

2023, Volume 8, Issue 11: 25409-25421. doi: 10.3934/math.20231296

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Research article

Blow-up to a shallow water wave model including the Degasperis-Procesi equation

Jin Hong ^{1
,
,},
Shaoyong Lai ²

1.
School of Mathematics and Statistics, Yili Normal University, Yili 835000, China
2.
School of Mathematics, Southwestern University of Finance and Economics, Chengdu 610030, China

Received: 04 June 2023 Revised: 31 July 2023 Accepted: 22 August 2023 Published: 31 August 2023
MSC : 35G25, 35L05

A nonlinear equation, depicting motions of shallow water waves and including the famous Degasperis-Procesi model, is considered. The key element is that we derive $ L^2 $ conservation law of solutions for the nonlinear equation, which leads to the bound of the solution itself. Using several estimates derived from the model, we obtain that when its solution blows up in the Sobolev space if and only if the space derivative of the solution tends to minus infinite.
- blow up,
- shallow water wave equation,
- generalized Degasperis-Procesi model,
- classical energy methods
Citation: Jin Hong, Shaoyong Lai. Blow-up to a shallow water wave model including the Degasperis-Procesi equation[J]. AIMS Mathematics, 2023, 8(11): 25409-25421. doi: 10.3934/math.20231296

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Abstract

A nonlinear equation, depicting motions of shallow water waves and including the famous Degasperis-Procesi model, is considered. The key element is that we derive $ L^2 $ conservation law of solutions for the nonlinear equation, which leads to the bound of the solution itself. Using several estimates derived from the model, we obtain that when its solution blows up in the Sobolev space if and only if the space derivative of the solution tends to minus infinite.

References

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