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Monatshefte für Mathematik

, Volume 188, Issue 1, pp 163–181 | Cite as

On blow-up criteria for a class of nonlinear dispersive wave equations with dissipation

  • Emil NovruzovEmail author
  • Betul Yazar
Article

Abstract

We study the Cauchy problem for the nonlinear dispersive wave equation with dissipative term on the real line, \(u_{t}-u_{xxt}+\left[ f\left( u\right) \right] _{x}-\left[ f\left( u\right) \right] _{xxx}+\left[ g\left( u\right) + \frac{f^{\prime \prime }\left( u\right) }{2}u_{x}^{2}\right] _{x}+\lambda \left( u-u_{xx}\right) =0\) that includes the corresponding dissipative version of the Camassa–Holm equation as well as the hyperelastic-rod wave equation with dissipative term (\(f(u)=ku^{2}/2\) and \(g(u)=\left( 3-k\right) u^{2}/2\)) as special cases. In the present paper we demonstrate the simple and new conditions on the initial data that lead to blow-up of the solution. In particular, we establish local in space criterion (i.e., a criterion involving only the properties of the data \(u_{0}\) in a neighborhood of a single point) which guarantees that solutions blow up in finite time.

Keywords

Dissipative rod equation Blow-up Shallow water 

Mathematics Subject Classification

35B44 35L05 37K10 

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Copyright information

© Springer-Verlag GmbH Austria 2017

Authors and Affiliations

  1. 1.Department of MathematicsGebze Technical UniversityGebzeTurkey

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