Weak solutions of linear equations of Sobolev type and semigroups of operators

We show that there is an infinitely differentiable semigroup for the equation $L\dot u=Mu$ if the $(p,\psi(\tau))$-condition introduced in this paper holds for the pair of operators $(L,M)$. In the case when the strong $(p,\psi(\tau))$-condition holds we have found the set of one-valued solubility of the weakened Cauchy problem for this equation. Our results supplement the theory of degenerate semigroups of operators and generalize in part the theorem on the generators of semigroups of class $(A)_\infty$ to the case of degenerate semigroups. We investigate the kernels and images of the semigroups constructed and consider various examples of operators for which the $(p,\psi(\tau))$-condition (the strong $(p,\psi(\tau))$-condition) holds.