Parabolic Boundary Value Problems in Cylindrical Domains
Samuil D. Eidelman, Nicolae V. Zhitarashu · Birkhäuser Basel eBooks · 1998
Let Ω+ = G × [0, ∞) be a semi-infinite cylinder with lateral surface S+ = г × [0, ∞); here G is a bounded domain in ℝ n (or the exterior of a bounded domain G: G = ℝ\G) with a smooth (C∞) (n - 1)-dimensional boundary г = ∂G. Suppose that in \({{\bar{\Omega }}_{ + }}\) there is given an operator L(x,t, D, D t ), uniformly parabolic in the sense of Petrovskiĭ, ord l ij = t j = 2bt’ j , and that on \({{\bar{S}}_{ + }}\) there is given a matrix boundary operator B(x,t, D, D t ), ord b qj = σq +t j , which satisfies the Lopаtinskiĭ condition on \({{\bar{S}}_{ + }}\) uniformly in (x’, t) ∈ S+. For simplicity, we assume that ord t b qj < t’ j and the coefficients of the operators l ij and b qj are infinitely smooth in x and t.