30 research outputs found
On the differentiability of weak solutions of an abstract evolution equation with a scalar type spectral operator on the real axis
Given the abstract evolution equation
with scalar type spectral operator in a complex Banach space, found are
conditions necessary and sufficient for all weak solutions of the equation,
which a priori need not be strongly differentiable, to be strongly infinite
differentiable on . The important case of the equation with a
normal operator in a complex Hilbert space is obtained immediately as a
particular case. Also, proved is the following inherent smoothness improvement
effect explaining why the case of the strong finite differentiability of the
weak solutions is superfluous: if every weak solution of the equation is
strongly differentiable at , then all of them are strongly infinite
differentiable on .Comment: A correction in Remarks 3.1, a few minor readability improvements.
arXiv admin note: substantial text overlap with arXiv:1707.09359,
arXiv:1706.08014, arXiv:1708.0506