Generalized correctness of the Cauchy problem and integrated semigroups (Q1363863)

The differential-operator Cauchy problem \[ u'(t)= Au(t),\quad t\geq 0,\quad u(0)= x, \] that is not uniformly correct in the Banach space \(X\) on the natural class of correctness \(-\text{dom}(A)\) has recently been a subject of great interest. Here, the authors show that the existence of a generalized solution in the distribution space \({\mathcal D}'(X)\) is equivalent to the existence of a locally integrated semigroup, while the existence of a generalized solution in the exponential distribution space \(S_\omega'(X)\) is equivalent to the existence of an exponentially bounded integrated semigroup. They also give an effective construction of the generalized solutions using the integrated semigroups, as well as an effective construction of the integrated semigroups using the generalized solutions.