Regularized semigroups, iterated Cauchy problems and equipartition of energy


The iterated Cauchy problem under consideration is {Mathematical expression} Here {A1,..., An} are unbounded linear operators on a Banach space. The initial value problem for (*) is governed by a semigroup of some sort. When each Ak is a (C0) semigroup generator, this semigroup is of class (C0) and was studied by J. T. Sandefur [26]. This result is extended to the case when each Ak generates a C-regularized semigroup (with C independent of k). This means one can solve u′=Au, u(0)=f∈C (Dom (A)) and get u(t)→0 whenever C-1f→0; here C is bounded and injective. When the Ak are commuting generator with Ak-Aj injective for k≠j, then the Goldstein-Sandefur d'Alembert formula [19] is extended, viz. solutions of (*) (with suitable restrictions on the initial data) are of the form {Mathematical expression} where ui is a solution of u′i=Aiui. Examples and applications are given. Included among the examples is the establishment of a form of equipartition of energy for the Laplace equation; equipartition of energy is wellknown for the wave equation. A final section of the paper deals with the absence of necessary conditions for equipartition of energy. © 1993 Springer-Verlag.

Publication Title

Monatshefte für Mathematik