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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 -stability, and asymptotic stability, of the -soliton of the Korteweg-de Vries (KdV) equation.
The -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 linked (together with the time derivative along the solutions) by direct simple relations to the eigenvalues, is used.
Let be one solution towith a non-homogeneous term , and , where is a bounded domain. We discuss an inverse problem of determining unknown functions by , after selecting input sources suitably, where is an arbitrary subboundary, denotes the normal derivative, and . In the case of , we prove the Lipschitz stability in the inverse problem if we choose from a set with an arbitrarily fixed subdomain . Moreover we can take by making special choices for , . The proof is...
Let y(h)(t,x) be one solution to
with a non-homogeneous term h, and ,
where is a bounded domain. We discuss an inverse problem
of determining n(n+1)/2 unknown functions aij by
,
after selecting input sources suitably, where is an arbitrary subboundary,
denotes the normal derivative, and
. In the case of , we prove
the Lipschitz stability in the inverse problem if we choose from a set with an arbitrarily fixed subdomain
. Moreover we can take
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 . This result was obtained in...
We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation
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
We then give conditions for the convergence, as , of the solution of the evolution equation to its stationary state.
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