Embedding Theorems in Banach-valued B-spaces and Maximal B-regular Differential-operator Equations

The embedding theorems in anisotropic Besov-Lions type spaces Bp,θ(R ;E0,E) are studied; here E0 and E are two Banach spaces. The most regular spaces Eα are found such that the mixed differential operators Dα are bounded from Bp,θ(R ;E0,E) to B q,θ(R ;Eα), where Eα are interpolation spaces between E0 and E depending on α= (α1,α2, . . . ,αn) and l = (l1, l2, . . . , ln). By using these results the separability of anisotropic differential-operator equations with dependent coefficients in principal part and the maximal B-regularity of parabolic Cauchy problem are obtained. In applications, the infinite systems of the quasielliptic partial differential equations and the parabolic Cauchy problems are studied.