Evolution equations for abstract differential operators (Q1856924)

The author considers evolution equations (EQs) of the form \[ \begin{aligned} u'(t) &= \sum_{|\mu|\leq m} a_\mu(t) A^\mu u(t)+ f(t),\quad 0\leq t\leq T,\\ u(0) &= x,\end{aligned} \] where \(A^\mu= A^{\mu_1}_1\cdots A^{\mu_n}_n\) and \(iA_j\) are commuting generators of bounded \(C_0\)-groups on a Banach space \(X\). Evolution families and regularized evolution families are constructed for strongly elliptic and some nonelliptic cases. Well-posedness and regularity of solutions of EQs are obtained. Applications to partial differential equations are given.