Blowup and ill-posedness results for a Dirac equation without gauge invariance

Mamoru Okamoto, Piero D'Ancona
2016 Evolution Equations and Control Theory  
We consider the Cauchy problem for a nonlinear Dirac equation on R n , n ≥ 1, with a power type, non gauge invariant nonlinearity ∼ |u| p . We prove several ill-posedness and blowup results for both large and small H s data. In particular we prove that: for (essentially arbitrary) large data in H n 2 + (R n ) the solution blows up in a finite time; for suitable large H s (R n ) data and s < n 2 − 1 p−1 no weak solution exist; when 1 < p < 1 + 1 n (or 1 < p < 1 + 2 n in n = 1, 2, 3), there exist
more » ... 2, 3), there exist arbitrarily small initial data data for which the solution blows up in a finite time.
doi:10.3934/eect.2016002 fatcat:igyzjrjq3zb3pbanu5g3rsqocq