Page 1

Displaying 1 – 13 of 13

Showing per page

L 2 -stability of multi-solitons

Claudio Muñoz (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

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.

L 2 -stability of the solutions to a nonlinear binary reaction-diffusion system of P.D.E.s

Salvatore Rionero (2005)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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.

Lipschitz stability in the determination of the principal part of a parabolic equation

Ganghua Yuan, Masahiro Yamamoto (2009)

ESAIM: Control, Optimisation and Calculus of Variations

Let y ( h ) ( t , x ) be one solution to t y ( t , x ) - i , j = 1 n j ( a i j ( x ) i y ( t , x ) ) = h ( t , x ) , 0 < t < T , x Ω with a non-homogeneous term h , and y | ( 0 , T ) × Ω = 0 , where Ω n is a bounded domain. We discuss an inverse problem of determining n ( n + 1 ) / 2 unknown functions a i j by { ν y ( h ) | ( 0 , T ) × Γ 0 , y ( h ) ( θ , · ) } 1 0 after selecting input sources h 1 , . . . , h 0 suitably, where Γ 0 is an arbitrary subboundary, ν denotes the normal derivative, 0 < θ < T and 0 . In the case of 0 = ( n + 1 ) 2 n / 2 , we prove the Lipschitz stability in the inverse problem if we choose ( h 1 , . . . , h 0 ) from a set { C 0 ( ( 0 , T ) × ω ) } 0 with an arbitrarily fixed subdomain ω Ω . Moreover we can take 0 = ( n + 3 ) n / 2 by making special choices for h , 1 0 . The proof is...

Lipschitz stability in the determination of the principal part of a parabolic equation

Ganghua Yuan, Masahiro Yamamoto (2008)

ESAIM: Control, Optimisation and Calculus of Variations

Let y(h)(t,x) be one solution to t y ( t , x ) - i , j = 1 n j ( a i j ( x ) i y ( t , x ) ) = h ( t , x ) , 0 < t < T , x Ω with a non-homogeneous term h, and y | ( 0 , T ) × Ω = 0 , where Ω n is a bounded domain. We discuss an inverse problem of determining n(n+1)/2 unknown functions aij by { ν y ( h ) | ( 0 , T ) × Γ 0 , y ( h ) ( θ , · ) } 1 0 after selecting input sources h 1 , . . . , h 0 suitably, where Γ 0 is an arbitrary subboundary, ν denotes the normal derivative, 0 < θ < T and 0 . In the case of 0 = ( n + 1 ) 2 n / 2 , we prove the Lipschitz stability in the inverse problem if we choose ( h 1 , . . . , h 0 ) from a set { C 0 ( ( 0 , T ) × ω ) } 0 with an arbitrarily fixed subdomain ω Ω . Moreover we can take 0 = ( n + 3 ) n / 2 by making special choices for...

Local attractivity in nonautonomous semilinear evolution equations

Joël Blot, Constantin Buşe, Philippe Cieutat (2014)

Nonautonomous Dynamical Systems

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...

Local center manifold for parabolic equations with infinite delay

Hana Petzeltová (1994)

Mathematica Bohemica

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.

Long-Time Asymptotics for the Navier-Stokes Equation in a Two-Dimensional Exterior Domain

Thierry Gallay (2012)

Journées Équations aux dérivées partielles

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 . This result was obtained in...

Long-time behavior of small solutions to quasilinear dissipative hyperbolic equations

Albert J. Milani, Hans Volkmer (2011)

Applications of Mathematics

We give sufficient conditions for the existence of global small solutions to the quasilinear dissipative hyperbolic equation u t t + 2 u t - a i j ( u t , u ) i 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 i j ( 0 , v ) i j v = h . We then give conditions for the convergence, as t , of the solution of the evolution equation to its stationary state.

Currently displaying 1 – 13 of 13

Page 1