For nonlinear heat and Schrödinger equations with non-algebraic power nonlinearities, the paper identifies the sharp Sobolev thresholds, s < p+2+1/q and s < p+5/2 respectively, with strong ill-posedness at the endpoint.
On the boundedness of the mapping f → |f | in Besov spaces
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On the optimal Sobolev threshold for evolution equations with rough nonlinearities
For nonlinear heat and Schrödinger equations with non-algebraic power nonlinearities, the paper identifies the sharp Sobolev thresholds, s < p+2+1/q and s < p+5/2 respectively, with strong ill-posedness at the endpoint.