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Blow-up of solutions to the DGH equation. (English) Zbl 1124.35079
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:
35Q53KdV-like (Korteweg-de Vries) equations
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35A20Analytic methods, singularities (PDE)
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
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References:
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