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.


Let $p>1$, $q>p$, $\lambda >0$ and $s\in ]1,p[$. We study, for $s\to p^{-}$, the behavior of positive solutions of the problem $-{\Delta }_{p}u=\lambda u^{s-1}+u^{q-1}$ in $\Omega$, $u_{\mid \partial \Omega }=0$. In particular, we give a positive answer to an open question formulated in a recent paper of the first author.

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.