On \(C_ 0\)-semigroups generated by elliptic second order differential expressions on \(L^ p\)-spaces. (English) Zbl 0852.47018

Liskevich, Vitali

Differ. Integral Equ. 9, No. 4, 811-826 (1996).

Summary: 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 \(C_0\)-semigroups. Lower order terms are considered as form-bounded perturbations of the generator of the symmetric sub-Markovian semigroup associated with the Dirichlet form. It is shown that the \(C_0\)-semigroup 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 semigroup constructed.

MSC:

47D06	One-parameter semigroups and linear evolution equations
47F05	General theory of partial differential operators
47D09	Operator sine and cosine functions and higher-order Cauchy problems
47D07	Markov semigroups and applications to diffusion processes

Keywords:

well-posedness in \(L^ p\); Cauchy problem; second-order parabolic equations; time-independent measurable coefficients; \(C_ 0\)-semigroups; form-bounded perturbations; generator; symmetric sub-Markovian semigroup; Dirichlet form; analyticity; \(L^ p\)-smoothness