High order regularity of the singularities of the solution of a parabolic equation in a singular domain

摘要

We are concerned, in this work, with the parabolic equation: (*) (partial deriv)/((partial deriv)t) u + (partial deriv)~4/((partial deriv)x)~4 u = f in the following non convex polygonal domain Ω described by the variables (t, x) ∈ R~2: Ω = (-1, 0) x (-1, 1) ∪ [0, 1) x (0, 1). The boundary conditions are of Cauchy-Dirichlet type while the second member f of the equation lies in the non symmetric Sobolev space H~(p,4p)(Ω) defined, for p ∈ N, by: H~(p,4p)(Ω) = {u ∈ L~2(Ω): (partial deriv)~(j+k)/(((partial deriv)t)~j((partial deriv)x)~k) u ∈ L~2(Ω), 4j + k ≤ 4p}. Here, L~2(Ω) denotes the well known Lebesgue space. If f satisfies some compatibility conditions, the "natural" space of solutions, in case of Ω regular enough (e.g., convex domain) is the Sobolev space P~(p+1,4(p+1))(Ω). According to some works, we know that the space of solutions may be different. In Sadallah (1996), we dealt with the equation (*) for p = 0. Now, we are interested in the optimal regularity of the singularities which appear in the solution when p is any integer. The main result of this work shows the existence of 2(p + 1) singularities (ν_(j,k))_(j = 0,1; k = 0,...,p) such that, for all f ∈ H~(p,4p)(Ω) satisfying some compatibility conditions, the solution belongs to the space H~(p+1,4(p+1))(Ω)⊕ ∑_(j = 0,1; k = 0,...,p)IRν_(j,k). In addition, the singularities fulfill the optimal regularity condition, for all j = 0, 1 and k = 0,...,p: ν_(j,k) ∈ H~(r_j +k, 4(r_j + k)) (Ω) < = > r_j (5 + 2j)/8.