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Steklov Mathematical Institute Seminar
January 17, 2013 16:00, Moscow, Steklov Mathematical Institute of RAS, Conference Hall (8 Gubkina)

Blow-up of smooth solutions to the Korteweg–de Vries equation

S. I. Pokhozhaev
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S. I. Pokhozhaev
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Abstract: By the method of the nonlinear capacity, proposed by S. I. Pokhozhaev in 1997, the blow-up problem for nonlinear partial differential equations of mathematical physics is studied. This method allows to consider the nonlinear systems and high-order equations for the fist time.
In particular, for Korteweg–de Vries equations and its modifications we obtain conditions on the smooth initial functions for the Cauchy problems and initial-boundary conditions for initial-boundary problems, for which smooth solutions blow-up in finite time. Also estimates for blow-up time are presented.
We demonstrate the examples, illustrated the mechanism and properties of the blow-up phenomena. In general, they are similar to the properties of solutions blow-up for the Cauchy problem for the nonlinear (cubic) Schrodinger equation in the three-dimensional space.

  1. S. I. Pohozaev, “Blow-up of smooth solutions of the Korteweg–de Vries equation”, Nonlinear Anal., 75:12 (2012), 4688–4698  crossref  mathscinet  zmath  isi  scopus
  2. S. I. Pokhozhaev, “Ob odnom klasse nachalno-kraevykh zadach dlya uravnenii tipa Kortevega–de Friza”, Differents. uravneniya, 48:3 (2012), 368–374  mathscinet  zmath; S. I. Pokhozhaev, “On a class of initial-boundary value problems for equations of Korteweg–de Vries type”, Differ. Equ., 48:3 (2012), 372–378  crossref  mathscinet  zmath  isi  scopus
  3. S. I. Pokhozhaev, “Ob otsutstvii globalnykh reshenii zadachi Koshi dlya uravneniya Kortevega–de Friza”, Funkts. analiz i ego pril., 46:4 (2012), 51–60  mathnet  crossref  zmath; S. I. Pokhozhaev, “On the nonexistence of global solutions of the Cauchy problem for the Korteweg–de Vries equation”, Funct. Anal. Appl., 46:4 (2012), 279–286  crossref  mathscinet  zmath  isi  scopus

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