Our aim in this paper is to study the asymptotic behavior, in terms of finite-dimensional attractors, of a sixth-order Cahn-Hilliard system. This system is based on a modification of the Ginzburg-Landau free energy proposed in [Torabi S., Lowengrub J., Voigt A., Wise S., A new phase-field model for strongly anisotropic systems, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 2009, 465(2105), 1337–1359], assuming isotropy.

The paper investigates the asymptotic behavior of a steady flow of an incompressible viscous fluid in a two-dimensional infinite pipe with slip boundary conditions and large flux. The convergence of the solutions to data at infinities is examined. The technique enables computing optimal factors of exponential decay at the outlet and inlet of the pipe which are unsymmetric for nonzero fluxes of the flow. As a corollary, the asymptotic structure of the solutions is obtained. The results show strong...

We construct global solutions to the Navier-Stokes equations with initial data small in a Besov space. Under additional assumptions, we show that they behave asymptotically like self-similar solutions.


For a fixed bounded open set $\Omega \subset {ℝ}^N$, a sequence of open sets ${\Omega }_n\subset \Omega$ and a sequence of sets ${\Gamma }_n\subset \partial \Omega \cap \partial {\Omega }_n$, we study the asymptotic behavior of the solution of a nonlinear elliptic system posed on ${\Omega }_n$, satisfying Neumann boundary conditions on ${\Gamma }_n$ and Dirichlet boundary conditions on $\partial {\Omega }_n\setminus {\Gamma }_n$. We obtain a representation of the limit problem which is stable by homogenization and we prove that this representation depends on ${\Omega }_n$ and ${\Gamma }_n$ locally.


We study the semi-classical asymptotic behavior as $(h\to 0)$ of scattering amplitudes for Schrödinger operators $-(1/2)h^2\Delta +V$. The asymptotic formula is obtained for energies fixed in a non-trapping energy range and also is applied to study the low energy behavior of scattering amplitudes for a certain class of slowly decreasing repulsive potentials without spherical symmetry.

We consider the Keller-Segel-Navier-Stokes system $$\left\{\begin{array}{cc}{n}_t+𝐮·\nabla n=\Delta n-\nabla ·\left(n\nabla v\right),\hfill & x\in \Omega ,t>0,\hfill \\ v_t+𝐮·\nabla v=\Delta v-v+w,\hfill & x\in \Omega ,t>0,\hfill \\ w_t+𝐮·\nabla w=\Delta w-w+n,\hfill & x\in \Omega ,t>0,\hfill \\ {𝐮}_t+(𝐮·\nabla )𝐮=\Delta 𝐮+\nabla P+n\nabla \phi ,\nabla ·𝐮=0,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$$ which is considered in bounded domain $\Omega \subset {ℝ}^N$$(N\in \{2,3\}...$

We study the existence of global in time and uniform decay of weak solutions to the initial-boundary value problem related to the dynamic behavior of evolution equation accounting for rotational inertial forces along with a linear nonlocal frictional damping arises in viscoelastic materials. By constructing appropriate Lyapunov functional, we show the solution converges to the equilibrium state polynomially in the energy space.