The aim of this note is to give a short review of our recent work (see [5]) with Miguel A. Alejo and Luis Vega, concerning the $L^2$-stability, and asymptotic stability, of the $N$-soliton of the Korteweg-de Vries (KdV) equation.

The $L^2$-stability (instability) of a binary nonlinear reaction diffusion system of P.D.E.s - either under Dirichlet or Neumann boundary data - is considered. Conditions allowing the reduction to a stability (instability) problem for a linear binary system of O.D.E.s are furnished. A peculiar Liapunov functional $V$ linked (together with the time derivative along the solutions) by direct simple relations to the eigenvalues, is used.

Let $y\left(h\right)(t,x)$ be one solution to$${\partial }_{t}y(t,x)-\sum _{i,j=1}^n{\partial }_j\left(a_{ij}\left(x\right){\partial }_{i}y(t,x)\right)=h(t,x),0\<t\<T,x\in \Omega$$with a non-homogeneous term $h$, and ${y|}_{(0,T)\times \partial \Omega }=0$, where $\Omega \subset {ℝ}^n$ is a bounded domain. We discuss an inverse problem of determining $n(n+1)/2$ unknown functions $a_{ij}$ by $\{{\partial }_{\nu }y\left(h_{\ell }\right){|}_{(0,T)\times {\Gamma }_0}$, $y\left(h_{\ell }\right)(\theta ,·){\}}_{1\le \ell \le {\ell }_0}$ after selecting input sources $h_1,...,h_{{\ell }_0}$ suitably, where ${\Gamma }_0$ is an arbitrary subboundary, ${\partial }_{\nu }$ denotes the normal derivative, $0\<\theta \<T$ and ${\ell }_0\in \mathbb{N}$. In the case of ${\ell }_0={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $(h_1,...,h_{{\ell }_0})$ from a set $\mathscr{H}\subset {\left\{C_0^{\infty }((0,T)\times \omega )\right\}}^{{\ell }_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$. Moreover we can take ${\ell }_0=(n+3)n/2$ by making special choices for $h_{\ell }$, $1\le \ell \le {\ell }_0$. The proof is...

Let y(h)(t,x) be one solution to $${\partial }_{t}y(t,x)-\sum _{i,j=1}^n{\partial }_j\left(a_{ij}\left(x\right){\partial }_{i}y(t,x)\right)=h(t,x),0<t<T,x\in \Omega$$ with a non-homogeneous term h, and ${y|}_{(0,T)\times \partial \Omega }=0$, where $\Omega \subset {ℝ}^n$ is a bounded domain. We discuss an inverse problem of determining n(n+1)/2 unknown functions aij by $\{{\partial }_{\nu }y\left(h_{\ell }\right){|}_{(0,T)\times {\Gamma }_0}$, $y\left(h_{\ell }\right)(\theta ,·){\}}_{1\le \ell \le {\ell }_0}$ after selecting input sources $h_1,...,h_{{\ell }_0}$ suitably, where ${\Gamma }_0$ is an arbitrary subboundary, ${\partial }_{\nu }$ denotes the normal derivative, $0<\theta <T$ and ${\ell }_0\in \mathbb{N}$. In the case of ${\ell }_0={(n+1)}^{2}n/2$, we prove the Lipschitz stability in the inverse problem if we choose $(h_1,...,h_{{\ell }_0})$ from a set $\mathscr{H}\subset {\left\{C_0^{\infty }((0,T)\times \omega )\right\}}^{{\ell }_0}$ with an arbitrarily fixed subdomain $\omega \subset \Omega$. Moreover we can take ${\ell }_0=(n+3)n/2$ by making special choices for...

We study the local attractivity of mild solutions of equations in the form u’(t) = A(t)u(t) + f (t, u(t)), where A(t) are (possible) unbounded linear operators in a Banach space and where f is a (possible) nonlinear mapping. Under conditions of exponential stability of the linear part, we establish the local attractivity of various kinds of mild solutions. To obtain these results we provide several results on the Nemytskii operators on the space of the functions which converge to zero at infinity...

The existence and attractivity of a local center manifold for fully nonlinear parabolic equation with infinite delay is proved with help of a solutions semigroup constructed on the space of initial conditions. The result is applied to the stability problem for a parabolic integrodifferential equation.

We study the long-time behavior of infinite-energy solutions to the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. The initial data we consider are finite-energy perturbations of a smooth vortex with small circulation at infinity, but are otherwise arbitrarily large. Using a logarithmic energy estimate and some interpolation arguments, we prove that the solution approaches a self-similar Oseen vortex as $t\to \infty$. This result was obtained in...

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation $$u_{tt}+2u_t-a_{ij}(u_t,\nabla u){\partial }_i{\partial }_{j}u=f$$ corresponding to initial values and source terms of sufficiently small size, as well as of small solutions to the corresponding stationary version, i.e. the quasilinear elliptic equation $$-a_{ij}(0,\nabla v){\partial }_i{\partial }_{j}v=h.$$ We then give conditions for the convergence, as $t\to \infty$, of the solution of the evolution equation to its stationary state.

We consider a 2D mathematical model describing the motion of a solution of surfactants submitted to a high shear stress in a Couette$-$Taylor system. We are interested in a stabilization process obtained thanks to the shear. We prove that, if the shear stress is large enough, there exists global in time solution for small initial data and that the solution of the linearized system (controlled by a nonconstant parameter) tends to 0 as $t$ goes to infinity. This explains rigorously some experiments.

We consider a 2D mathematical model describing the motion of a solution of surfactants submitted to a high shear stress in a Couette-Taylor system. We are interested in a stabilization process obtained thanks to the shear. We prove that, if the shear stress is large enough, there exists global in time solution for small initial data and that the solution of the linearized system (controlled by a nonconstant parameter) tends to 0 as t goes to infinity. This explains rigorously some experiments. ...

A model of a cable stayed bridge is proposed. This model describes the behaviour of the center span, the part between pylons, hung on one row of cable stays. The existence, the uniqueness of a solution of a time independent problem and the continuous dependence on data are proved. The existence and the uniqueness of a solution of a linearized dynamic problem are proved. A homogenizing procedure making it possible to replace cables by a continuous system is proposed. A nonlinear dynamic problem connected...