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

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.