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Of current mathematical and physical interest are equatorial waves, in the setting of an azimuthal two-dimensional inviscid flow in a two-layer fluid with a flat bed and a free surface. The Hamiltonian formulation for nonlinear governing equations is adequate for structure-preserving perturbations, at the linear and at the nonlinear level have been derived. Since the ocean energy is concentrated in the long-wave propagation modes, it is of interest to pursue an in-depth analysis of long waves. In the present paper we will consider the process in the absence of vorticity.

We investigate the blow-up phenomena for the two-component generalization of hyper-elastic rod equation on the real line. We establish some blow-up criteria for system of coupled equations with den...

For s 3 / 2 , it is known (see [24] ) that the Cauchy problem for the b-family of equations is ill-posed in Sobolev spaces H s when b > 1 . The proof of ill-posedness depends naturally on the value of b, and is based on the construction of peakon–antipeakon solutions with interesting properties which allows to make conclusion on ill-posedness. In this context the construction of such type of the solution for b 1 is very attractive problem which help shed light on the ill-posedness problem in this case. Thus, in the present paper we consider the ill-posedness of the b-family of equations with additional term for insufficiently investigated case b 1 on the line.

We investigate the blow-up phenomena for the two-component generalizations of Camassa-Holm equation on the real line. We establish some a local-in-space blow-up criterion for system of coupled equations under certain natural initial profiles. Presented result extends and specifies the earlier blow-up criteria for such type systems.

This paper is concerned with blow-up phenomena for the nonlinear dispersive wave equation on the real line, u t − u x x t + [ f ( u ) ] x − [ f ( u ) ] x x x + [ g ( u ) + f ″ ( u ) 2 u x 2 ] x = 0 that includes the Camassa–Holm equation as well as the hyperelastic-rod wave equation ( f ( u ) = k u 2 / 2 and g ( u ) = ( 3 − k ) u 2 / 2 ) as special cases. We establish some a local-in-space blow-up criterion (i.e., a criterion involving only the properties of the data u 0 in a neighborhood of a single point) simplifying and precising earlier blow-up criteria for this equation.

We study the Cauchy problem for the weakly dissipative Dullin–Gottwald–Holm equation describing unidirectional propagation of surface waves in a shallow water regime: u t − α 2 u x x t + c 0 u x + 3 u u x + γ u x x x + λ ( u − α 2 u x x ) = α 2 ( 2 u x u x x + u u x x x ) . In the present paper we demonstrate the simple conditions on the initial data that lead to blow-up of the solution 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.

The subject of this investigation is the qualitative properties of positive solutions of the following nonhomogeneous equation or, after the change v = u σ, of equation in unbounded domain R + × RN , where the term g(s) is supposed to satisfy just a lower polynomial growth condition and g′(s) > −l 1. The existence of the – global attractor for the equation (1′) is proved.

In this paper we discuss the blow-up property of the positive solutions of the mixed problem for a one-dimensional polytropic filtration equation with source.

The subject of this investigation is the qualitative properties of solutions of the equation where nonlinear term g is supposed to satisfy the upper polynomial growth condition of arbitrary order and g′(s) > −l. The existence of a , -global attractor for this equation with Dirichlet boundary condition is proved.

The subject of this investigation is the blow-up phenomena of the positive solutions of the mixed problem for the one-dimensional polytropic filtration equation with inhomogeneous density and source. It is shown that under certain conditions on the nonlinearities and data, blow-up will occur at some finite time. Note that the technique applied for the proof does not use the Zel'dovich-Kompaneets-Barenblatt solutions, since the construction of such type of function is more complicated in our case. Therefore, we obtain a result by multiplying on a special factor which has convenient properties. In particular, by choosing the parameters of the factor and using the properties of the solution, we obtain the inequality which allows us to show the blow up phenomena. (C) 2015 AIP Publishing LLC.

In this paper, we consider the dissipative Camassa-Holm equation with arbitrary dispersion coefficient and compactly supported initial data. We demonstrate the simple conditions on the initial data that lead to finite time blow-up of the solution in finite time or guarantee that the solution exists globally. Also, propagation speed for the equation under consideration is investigated. (C) 2014 Elsevier Inc. All rights reserved.

For a rapidly spatially oscillating nonlinearity 𝑔 we compare solutions 𝑢 𝜖 of non-Newtonian filtration equation 𝜕 𝑡 𝛽 ( 𝑢 𝜖 ) − 𝐷 ( | 𝐷 𝑢 𝜖 | 𝑝 − 2 𝐷 𝑢 𝜖 + 𝜑 ( 𝑢 𝜖 ) 𝐷 𝑢 𝜖 ) + 𝑔 ( 𝑥 , 𝑥 / 𝜖 , 𝑢 𝜖 ) = 𝑓 ( 𝑥 , 𝑥 / 𝜖 ) with solutions 𝑢 0 of the homogenized, spatially averaged equation 𝜕 𝑡 𝛽 ( 𝑢 0 ) − 𝐷 ( | 𝐷 𝑢 0 | 𝑝 − 2 𝐷 𝑢 0 + 𝜑 ( 𝑢 0 ) 𝐷 𝑢 0 ) + 𝑔 0 ( 𝑥 , 𝑢 0 ) = 𝑓 0 ( 𝑥 ) . Based on an 𝜀 -independent a priori estimate, we prove that | | 𝛽 ( 𝑢 𝜖 ) − 𝛽 ( 𝑢 0 ) | | 𝐿 1 ( Ω ) ≤ 𝐶 𝜖 𝑒 𝜌 𝑡 uniformly for all 𝑡 ≥ 0 . Besides, we give explicit estimate for the distance between the nonhomogenized 𝐴 𝜖 and the homogenized 𝐴 0 attractors in terms of the parameter 𝜖 ; precisely, we show fractional-order semicontinuity of the global attractors for 𝜖 ↘ 0 ∶ d i s t 𝐿 1 ( Ω ) ( 𝐴 𝜖 , 𝐴 0 ) ≤ 𝐶 𝜖 𝛾 .

In this paper, we study the Cauchy problem for the weakly dissipative Dullin–Gottwald–Holm equation. We establish certain conditions on the initial datum to guarantee that the corresponding positive strong solutions blow up in finite time.