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We use the Hamiltonian structure of equatorial wave-current interactions in the case of non-zero constant vorticity in the near-surface homogeneous layer, studying the dispersion relation for linear long waves outrunning the currents (downstream waves). We also investigate the behavior of the lifespan of the solution of the evolution equation for nonlinear long internal waves with respect to vorticity values.

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.

The subject of this investigation is the long time behavior of the positive solutions of the following nonhomogeneous equation $$\begin{aligned} \rho \left( \left| x\right| \right) \frac{\partial \beta \left( v\right) }{\partial t}-\overset{N}{\underset{i=1}{\sum }}D_{i}(|D_{i}v|^{ \lambda -1}D_{i}v)+\rho \left( \left| x\right| \right) g\left( \beta \left( v\right) \right) +l\beta ^{1+m}\left( v\right) =f\left( x\right) \qquad \end{aligned}$$ (1) in unbounded domain \( \mathbb {R} _{+}\times \mathbb {R} ^{N},\) where the term \(g\left( s\right) \) is supposed to satisfy a condition \(g^{\prime }\left( s\right) >-l_{1}\) and \(D_{i}=\partial _{x_{i}}\) . The existence of the global attractor for the Eq. (1) in \(L^{1+\theta }\left( \mathbb {R} ^{N},\rho \right) =\left\{ v;v\rho ^{1/(1+\theta )}\in L^{1+\theta }\left( \mathbb {R} ^{N}\right) \right\} \) is proved.

We study the Cauchy problem for the weakly dissipative Camassa–Holm equation: $$u_{t}-u_{txx}+3uu_{x}+\lambda \left( u-u_{xx} \right)=2u_{x}u_{xx}+uu_{xxx}, \quad t >0 ,x\in \mathbb{R}$$ . A new blow-up result for positive strong solutions of the equation with certain profiles is presented. In particular, we use a condition where the initial data u 0 and its derivative u 0x are not simultaneously involved and the parameter λ is not bounded from above.