On the strong smoothness of weak solutions of an abstract evolution equation i. differentiability

Abstract

For the evolution equation y' ^′(t)=Ay(t) with a normal operator A in a Hilbert space, conditions on A are found which are necessary and sufficient for all weak solutions of the equation to be strongly differentiable. Certain effects of smoothness improvement of the weak solutions are analyzed. The strong infinite differentiability of weak solutions of the equation with a symmetric operator is proved.

Keywords:

• Weak solution
• normal operator

AMS:

• 34G10
• 47B155
• 47D06