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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

u t -α 2 u txx +c 0 u x +3uu x +γu xxx =α 2 (2u x u xx +uu xxx ),x,t>0,u(x,t=0)=u 0 (x),x·

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