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Results of nonexistence of solutions for some nonlinear evolution problems

Medjahed Djilali, Ali Hakem (2019)

Commentationes Mathematicae Universitatis Carolinae

In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), u t t + f ( x ) u t + ( - Δ ) α / 2 ( u m ) = h ( t , x ) | u | p , posed in ( 0 , T ) × ℝ N , where ( - Δ ) α / 2 , 0 < α ≤ 2 is α / 2 -fractional power of - Δ . Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a 2 × 2 system of the same type.

Robust a priori error analysis for the approximation of degree-one Ginzburg-Landau vortices

Sören Bartels (2005)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

This article discusses the numerical approximation of time dependent Ginzburg-Landau equations. Optimal error estimates which are robust with respect to a large Ginzburg-Landau parameter are established for a semi-discrete in time and a fully discrete approximation scheme. The proofs rely on an asymptotic expansion of the exact solution and a stability result for degree-one Ginzburg-Landau vortices. The error bounds prove that degree-one vortices can be approximated robustly while unstable higher...

Robust a priori error analysis for the approximation of degree-one Ginzburg-Landau vortices

Sören Bartels (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

This article discusses the numerical approximation of time dependent Ginzburg-Landau equations. Optimal error estimates which are robust with respect to a large Ginzburg-Landau parameter are established for a semi-discrete in time and a fully discrete approximation scheme. The proofs rely on an asymptotic expansion of the exact solution and a stability result for degree-one Ginzburg-Landau vortices. The error bounds prove that degree-one vortices can be approximated robustly while unstable higher...

Scattering amplitude for the Schrödinger equation with strong magnetic field

Laurent Michel (2005)

Journées Équations aux dérivées partielles

In this note, we study the scattering amplitude for the Schrödinger equation with constant magnetic field. We consider the case where the strengh of the magnetic field goes to infinity and we discuss the competition between the magnetic and the electrostatic effects.

Scattering on stratified media: the microlocal properties of the scattering matrix and recovering asymptotics of perturbations

Tanya Christiansen, M. S. Joshi (2003)

Annales de l’institut Fourier

The scattering matrix is defined on a perturbed stratified medium. For a class of perturbations, its main part at fixed energy is a Fourier integral operator on the sphere at infinity. Proving this is facilitated by developing a refined limiting absorption principle. The symbol of the scattering matrix determines the asymptotics of a large class of perturbations.

Second order quasilinear functional evolution equations

László Simon (2015)

Mathematica Bohemica

We consider second order quasilinear evolution equations where also the main part contains functional dependence on the unknown function. First, existence of solutions in ( 0 , T ) is proved and examples satisfying the assumptions of the existence theorem are formulated. Then a uniqueness theorem is proved. Finally, existence and some qualitative properties of the solutions in ( 0 , ∞ ) (boundedness and stabilization as t → ∞ ) are shown.

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