Cahiers du CEREMADE 

Unité
Mixte de Recherche du C.N.R.S. N°7534 

Abstract : We prove that for a Dirac operator with no resonance at thresholds
nor eigenvalue at thresholds the propagator satisfies propagation
and dispersive estimates.
When this linear operator has only two simple eigenvalues close
enough, we study an associated class of nonlinear Dirac equations
which have stationary solutions. As an application of our decay
estimates, we show that these solutions have stable directions which
are tangent to the subspaces associated with the continuous spectrum
of the Dirac operator. This result is the analogue, in the Dirac
case, of a theorem by Tsai and Yau about the SchrÃ¶dinger equation.
To our knowledge, the present work is the first mathematical study
of the stability problem for a nonlinear Dirac equation. 





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