Evolution equations as space-time operator equations (Q2367599)

The author presents the transition from the classical semigroup theory for the initial value problem \[ \dot u(t)= 2\pi i Au(t)+ f(t), \quad t>0, \qquad u(0)= u_ 0\in Y_ 1\equiv D_ 0 (A) \] with \(f\in C_ 1 (\overline {\mathbb{R}}_ +, y)\) to the so-called ``space-time evolution problem'' \[ ((D_ \nu -A)- i\nu) u= (2\pi i)^{-1} (f+ \text{sgn} (\nu) \delta\otimes u_ 0)\equiv g, \] where \(D_ \nu: {\overset \circ C}_ \infty (\mathbb{R}) \subseteq H^ \nu\to H^ \nu\), \(\varphi \mapsto (2\pi i)^{-1} \dot\varphi+ i\nu\varphi\) introduced in connection with the Laplace transform [see the author: `A Hilbert space approach to some classical transforms' (1989; Zbl 0753.44002)] and \(D_ \nu -A\) is an essentially normal operator on the family of spaces \({\mathcal H}= (H^ \nu_ k \otimes Y_ j )_{k,j}\) and \({\mathcal Y}= (Y_ j)_ j\) is the scale associated with \(A\). This transition is the same as the transition from the classical elliptic differential equation theory to the Sobolev space theory. Then the author presents the regularity theory for ``space-time evolution problems'' and gives its applications to the anisotropic heat equation and Schrödinger equation.