  We consider a nonlinear evolution problem with an asymptotic parameter and
construct examples in which the linearized operator has spectrum uniformly
bounded away from Re z >= 0 (that is, the problem is spectrally stable), yet
the nonlinear evolution blows up in short times for arbitrarily small initial
data.
  We interpret the results in terms of semiclassical pseudospectrum of the
linearized operator: despite having the spectrum in Re z < -c < 0, the
resolvent of the linearized operator grows very quickly in parts of the region
Re z > 0. We also illustrate the results numerically.
