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Zbl 1124.35079
Zhou, Yong
Blow-up of solutions to the DGH equation.
(English)
[J] J. Funct. Anal. 250, No. 1, 227-248 (2007). ISSN 0022-1236

Summary: Firstly we find best constants for two convolution problems on the unit circle via a variational method. Then we apply the best constants on a nonlinear integrable shallow water equation the Dullin-Gottwald-Holm equation \align & u_t-\alpha^2u_{txx}+c_0u_x+3uu_x+\gamma u_{xxx}=\alpha^2(2u_xu_{xx}+uu_{xxx}),\ x\in\bbfR,\ t>0,\\ & u(x,t=0)=u_0(x),x\in\bbfR.\endalign to give sufficient conditions on the initial data, which guarantee finite time singularity formation for the corresponding solutions. Finally, we discuss the blow-up phenomena for the nonperiodic case.
MSC 2000:
*35Q53 KdV-like equations
76B15 Wave motions (fluid mechanics)
35A20 Analytic methods (PDE)
37K10 Completely integrable systems etc.

Keywords: best constant; convolution problem; integrable equation; singularity; shallow water equation

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