Norm continuity (for \(t>0\)) of propagators of arbitrary order abstract differential equations in Hilbert spaces
The authors deal with the abstract Cauchy problem \[ u^{(n)} (t)+ \sum^{n-1}_{k=0} A_ku^{(k)} (t)=0,\;t\geq 0,\;u^{(k)} (0)=u_k\;(0\leq k\leq n-1), \] where \(A_k\) are densely defined closed linear operators on a complex Hilbert space. Assuming that the problem is wellposed let \(S_0\), \(S_1, \dots, S_{n-1}\) be the mappings defined by \(S_k(t)x =u_k(t)\), \(t\geq 0\) where \(u_k (\cdot)\) is the solution with \(u_k^{(\ell)} (0)= \delta_{k\ell} x\). Then a necessary and sufficient condition for the mappings \(S_\ell\) to be norm continuous is given. An illustrative example closes the paper.
scientific article; zbMATH DE number 1005488