Abstract
A class of solved with respect to the \(m\) th order derivative evolution linear equations in Banach spaces on the real axis is considered. The equation is not endowed by initial conditions. A theorem on criterion in terms of the Fourier transform for analytic in a containing \(\mathbb{R}\) bisectorial region functions with values in a Banach space is proved. On the basis of this criterion we define a class of bisectorial operators. For the equation with a bisectorial operator, it is proved a unique solution existence. Moreover, it is shown that the unique solution has the form of the convolution of the inverse Fourier transform for \(((-i\omega)^{m}{-}A)^{-1}\) and of the right-hand side of the equation. The obtained general results are used in the study of a boundary value problem to a class of partial differential equations.
