A condition for generalized solutions of a parabolic problem for a Petrovskii system to be classical

We obtain a new sufficient condition under which generalized solutions to a parabolic initial-boundary-value problem for a Petrovskii system and the homogeneous Cauchy data are classical. The condition is formulated in terms of the belonging of the right-hand sides of the problem to some anisotropic H\"ormander spaces.


Introduction
In the theory of partial differential equations, of great importance are explicit conditions that guarantee a required regularity of solutions to the equations under study. As a rule, initial-boundary-value problems are investigated on appropriate pairs of normed distribution spaces. Once a solvability result for a problem is obtained, it is natural to ask when a distributional (i.e. generalized) solution to this problem is classical. In other words, when one may calculate the left-hand sides of the problem via classical partial derivatives and via traces of continuous functions. An answer to this question is usually given in terms of the belonging of the right-hand sides of the problem to relevant distribution spaces. The more finely a scale of these spaces is calibrated, the more exact conditions will be got.
Investigating parabolic problems, one usually uses anisotropic Sobolev spaces or Hölder spaces parametrized with a pair of real numbers [1,5,7,18,22]. To obtain more precise results about regularity of solutions of these problems, it is natural to resort to distribution spaces characterized with the help of function parameters. Broad classes of such spaces were introduced and investigated by Hörmander [2, Section 2.2] and Volevich and Paneah [23]. Of late years a theory of solvability of general parabolic problems is developed for some Hilbert anisotropic Hörmander spaces [8,9,[12][13][14][15][16][17]. Their order of regularity is given by a pair of real numbers s and s/(2b), where 2b is the parabolic weight of the problem, and by a function ϕ : [1, ∞) → (0, ∞) that varies regularly at infinity. The function parameter ϕ defines additional (positive or negative) regularity of distributions forming these spaces. The ϕ(·) ≡ 1 case provides the anisotropic Sobolev space of order s with respect to the spatial variables and order s/(2b) with respect to the time variable. A core of this theory consists of isomorphism theorems for operators induced by parabolic problems and acting between appropriate Hörmander spaces. (Somewhat earlier a theory of elliptic boundary-value problems was built for isotropic versions of these spaces [19,20].) The present paper investigates a parabolic initial-boundary-value problem for a Petrovskii parabolic system and the homogeneous Cauchy data. An isomorphism theorem for operators generated by this problem on pairs of appropriate inner product Hörmander spaces is proved in [12]. The purpose of the present paper is to supplement this theorem with a corresponding sufficient condition for generalized solutions of the parabolic problem to be classical in the sense mentioned above. The condition is formulated in terms of the belonging of the right-hand sides of the problem to suitable anisotropic Hörmander spaces. The use of the function parameter ϕ allows us to achieve the minimal admissible value of the number parameter s, which is not possible in the framework of Sobolev spaces or Hölder spaces [3,4,22]. As to scalar parabolic problems, conditions of this type are obtained in [10,11].
We consider the following parabolic initial-boundary-value problem in Ω: for all x ∈ G, t ∈ (0, τ ) and j ∈ {1, . . . , N }; for all x ∈ Γ, 0 < t < τ and j ∈ {1, . . . , m}; Note that the initial data (2.3) are assumed to be zero. The linear partial differential operators (PDOs) used in the problem are of the form for all admissible values of the indexes j and k. Here, the positive integers N ≥ 2, b, and κ 1 , . . . , κ N are arbitrarily chosen; m := b(κ 1 + · · · + κ N ), and l 1 , . . . , l m ∈ Z. The even number 2b is called the parabolic weight of this problem. All coefficients of the PDOs A j,k and B j,k are supposed to be infinitely smooth complex-valued functions given on Ω and S respectively; i.e., each . We use the notation D α x := D α1 1 . . . D αn n , with D k := i ∂/∂x k , and ∂ t := ∂/∂t for the partial derivatives of functions depending on x = (x 1 , . . . , x n ) ∈ R n and t ∈ R. Here, i is imaginary unit, and α = (α 1 , ..., α n ) is a multi-index, with |α| := α 1 + · · · + α n . In formulas (2.4) and (2.5) and their analogs, we take summation over the integer-valued nonnegative indices α 1 , ..., α n and β that satisfy the condition written under the integral sign.
We assume that the initial-boundary value problem (2.1)-(2.3) is Petrovskii parabolic in the cylinder Ω. Let us recall the corresponding definition [22, Section 1, § 1]. Define the principal symbols of the PDOs (2.4) and (2.5) as follows: These symbols are homogeneous polynomials in ξ := (ξ 1 , . . . , ξ n ) ∈ C n and p ∈ C jointly (as usual, ξ α := ξ α1 1 . . . ξ αn n ). Consider the matrices The problem (2.1)-(2.3) is said to be Petrovskii parabolic in Ω if it satisfies the following three conditions: (i) For arbitrary points x ∈ G and t ∈ [0, τ ] and every vector ξ ∈ R n , all the roots p(x, t, ξ) of the polynomial det A (0) (x, t, ξ, p) in p ∈ C satisfy the inequality Re p(x, t, ξ) ≤ −δ |ξ| 2b for some number δ > 0 that does not depend on x, t, and ξ. (ii) Each equation in the system (2.1) is solvable with respect to the derivative ∂ κj t u j , where j is the number of this equation, and does not contain any derivative of the form ∂ κ k t u k where k = j. Thus, we may and do assume that a (0,0,...,0),κ k j,k (x, t) ≡ δ j,k whenever j, k ∈ {1, . . . , N } (as usual, δ j,k is the Kronecker delta).
To formulate the third condition, we fix a number δ 1 ∈ (0, δ), where δ has appeared in Condition (i), and then arbitrarily choose a point x ∈ Γ, real number t ∈ [0, τ ], vector ξ ∈ R n tangent to the boundary Γ at x, and number p ∈ C such that Re p ≥ −δ 1 |ξ| 2b and |ξ| + |p| = 0. Let ν(x) denote the unit vector of the inward normal to Γ at x. It follows from Condition (i) and the inequality n ≥ 2 that the polynomial det A (0) (x, t, ξ+ζν(x), p) in ζ ∈ C has m roots ζ + j (x, t, ξ, p), j = 1, . . . , m, with positive imaginary part and m roots with negative imaginary part provided that each root is taken the number of times equal to its multiplicity.
The third condition is formulated as follows: (iii) For each positive number δ 1 < δ and for every choice of the parameters x, t, ξ p indicated above, the rows of the matrix are linearly independent modulo the polynomial m j=1 (ζ − ζ + j (x, t, ξ, p)). Here, A (0) is the transposed matrix of the cofactors of entries of A (0) .
Note that Conditions (i) and (ii) means that the system (2.1) is uniformly 2b-parabolic in the sense of Petrovskii in Ω [21], whereas Condition (iii) claims that the collection of boundary conditions (2.2) covers the parabolic system (2.1) on S.
It is a special case of the spaces B p,µ introduced by Hörmander [2, Section 2.2]; namely, H s,sγ;ϕ (R k+1 ) = B p,µ provided that p = 2 and µ(ξ, η) ≡ r s γ (ξ, η)ϕ(r γ (ξ, η)). If ϕ(·) ≡ 1, the space H s,sγ;ϕ (R k+1 ) becomes the anisotropic Sobolev space H s,sγ (R k+1 ). Generally, we have the dense continuous embeddings Basing on H s,sγ;ϕ (R k+1 ), consider some Hilbert function spaces relating to the problem with u ∈ H s,sγ;ϕ + (V ). This space is Hilbert and separable with respect to this norm. Specifically, the set We also consider the space H s,sγ;ϕ + (S) on the lateral boundary S of the cylinder Ω, we restricting ourselves to the s > 0 case. Briefly saying, this space consists of all functions v ∈ L 2 (S) that yield functions from the space H s,sγ;ϕ + (Π) on Π := R n−1 × (0, τ ) with the help of some local coordinates on S. Let us turn to a detailed definition.
If ϕ(·) ≡ 1, we will omit the index ϕ in the designations of the spaces considered in this section.
The use of Hörmander spaces allows us to attain the minimal admissible values of the number parameters in conditions (4.1) and (4.2). If we formulate an analog of this theorem using anisotropic Sobolev spaces (i.e. restricting ourselves to the case where ϕ 1 = ϕ 2 = ϕ 3 = 1), we have to claim that the right-hand sides of the problem under investigation satisfy these conditions for certain σ 1 > b + n/2, σ 2 > l 0 + b + n/2, and σ 3 > −b + n/2.

Proof of the main result
The proof is based on the following regularity property of the generalized solutions to the considered problem [12, Theorem 3]: for σ := p + b + n/2 − 2bκ k and some function parameter ϕ ∈ M subject to Then the generalized derivatives D α x ∂ β t u k (x, t) of the component u k (x, t) of u are continuous on Ω 0 ∪ Ω ′ whenever 0 ≤ |α| + 2bβ ≤ p.
Theorem 4.1 is proved. Bibliography