On $C_0$-semigroups generated by elliptic second order differential expressions on $L^p$-spaces

Abstract

We study well-posedness in $L^P$ of the Cauchy problem for second order parabolic equations with time-independent measurable coefficients by means of constructing corresponding Cosernigroups. Lower order terms are considered as form-bounded perturbations of the generator of the symmetric submarkovian sernigroup associated with the Dirichlet form. It is shown that the Cosernigroup corresponding to the Cauchy problem exists in a certain interval in the scale of $L^P$-spaces which depends only on form-bounds of perturbations. We establish also analyticity and $L^P$ -smoothness of the sernigroup constructed.