Alp Eden

Global existence and nonexistence results for a generalized Davey-Stewartson system

We consider a system of three equations, which will be called generalized Davey–Stewartson equations, involving three coupled equations, two for the long waves and one for the short wave propagating in an infinite elastic medium. We classify the system according to the signs of the parameters. Conserved quantities related to mass, momentum and energy are derived as well as a specific instance of the so-called virial theorem. Using these conservation laws and the virial theorem both global existence and nonexistence results are established under different constraints on the parameters in the elliptic–elliptic–elliptic case.

Standing waves for a generalized Davey–Stewartson system

In this paper, we establish the existence of non-trivial solutions for a semi-linear elliptic partial differential equation with a non-local term. This result allows us to prove the existence of standing wave (ground state) solutions for a generalized Davey–Stewartson system. A sharp upper bound is also obtained on the size of the initial values for which solutions exist globally.

Finite-dimensional attractors for a general class of nonautonomous wave equations

Abstract Our aim in this note is to construct attractors and exponential attractors for a general class of nonautonomous semilinear wave equations. Following the approach described in [1], we define a semigroup S(t) associated to an autonomous system, and then prove, using an energy functional, that S(t) is an α-contraction and satisfies the squeezing property.

Global solvability and blow up for the convective Cahn-Hilliard equations with concave potentials

We study initial boundary value problems for the unstable convective Cahn-Hilliard (CH) equation, i.e., the Cahn Hilliard equation whose energy integral is not bounded below. It is well-known that without the convective term, the solutions of the unstable CH equation ∂tu+∂x4u+∂x2(|u|pu)=0 may blow up in finite time for any p > 0. In contrast to that, we show that the presence of the convective term u∂xu in the Cahn-Hilliard equation prevents blow up at least for 0 0. In contrast to that, we show that the presence of the convective term u∂xu in the Cahn-Hilliard equation prevents blow up at least for 0<p<49. We also show that the blowing up solutions still exist if p is large enough (p ⩾ 2). The related equations like Kolmogorov-Sivashinsky-Spiegel equation, sixth order convective Cahn-Hilliard equation, are also considered.

A note on the global existence of small amplitude solutions to a generalized Davey–Stewartson system

In this paper, we are interested in the Cauchy problem for a generalized Davey–Stewartson (GDS) system. We establish the global time existence of small mass solutions for the GDS system in the elliptic–hyperbolic–hyperbolic case.

On the existence of special solutions of the generalized Davey–Stewartson system

Abstract In this note we show that for certain choice of parameters the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system admits time-dependent travelling wave solutions of the kind given in [V.A. Arkadiev, A.K. Pogrebkov, M.C. Polivanov, Inverse scattering transform method and soliton solutions for Davey–Stewartson II equation, Physica D 36 (1989) 189–197] for the hyperbolic Davey–Stewartson system. These solutions lead to radial solutions as well. We also find the sufficient conditions for non-existence of travelling wave solutions for the hyperbolic–elliptic–elliptic generalized Davey–Stewartson system by using the point of view developed in [A. Eden, T.B. Gurel, E. Kuz, Focusing and defocusing cases of the purely elliptic generalized Davey–Stewartson system, IMA J. Appl. Math. (in press)].