Initial boundary value problems for second order parabolic systems in cylinders with polyhedral base

The purpose of this article is to establish the well posedness and the regularity of the solution of the initial boundary value problem with Dirichlet boundary conditions for second-order parabolic systems in cylinders with polyhedral base.

Boundary value problems for partial differential equations and systems in nonsmooth domains have been attracted attentions of many mathematicians for more than last 50 years. We are concerned with initial boundary value problems (IBVP) for parabolic equations and systems in nonsmooth domains. These problems in cylinders with bases containing conical points have been investigated in [1,2] in which the regularity and the asymptotic behaviour near conical points of the solutions are established. Parabolic equations with discontinuous coefficients in Lipschitz domains have also been studied (see [3] and references therein).

In this study, we consider IBVP with Dirichlet boundary conditions for second-order parabolic systems in both cases of finite cylinders and infinite cylinders whose bases are polyhedral domains. Firstly, we prove the well posedness of this problem by Galerkin's approximating method. Next, by this method we obtain the regularity in time of the solution. Finally, we apply the results for elliptic boundary value problems in polyhedral domains given in [4,5] and former our results to deal with the global regularity of the solution.

Let Ω be an open polyhedral domain in ℝⁿ (n = 2, 3), and 0 < T ≤ ∞. Set Q_T = Ω × (0, T), S_T = ∂Ω × (0, T). For a vector-valued function u = (u₁, u₂, ..., u_s) and p = (p₁, p₂, ..., p_n) ∈ ℕⁿ we use the notation .

Let m, k be non negative integers. We denote by H^m(Ω), the usual Sobolev spaces as in [6]. By the notation (., .) we mean the inner product in L₂(Ω).

We denote by H^m,k(Q_T, γ) (γ ∈ ℝ) the weighed Sobolev space of vector-valued functions u defined in Q_T with the norm

Let us note that if T < +∞, then H^m,k(Q_T, γ) ≡ H^m,k(Q_T).

The space is the closure in H^m,k(Q_T, γ) of the set consisting of all vector-valued functions u ∈ C^∞(Q_T) which vanish near S_T.

Let ∂_singΩ be the set of all singular points of ∂Ω, namely, the set of vertexes of Ω for the case n = 2 and the union of all edges of Ω for the case n = 3. Let ρ(x) be the distance from a point x ∈ Ω to the set ∂_singΩ. For a ∈ ℝ, we denote by the weighed Sobolev space of vector functions u defined on Ω with the norm

It is obvious from the definition that continuous imbeddings hold for all a ≤ 1.

The weighed Sobolev spaces are defined as sets of all vector-valued functions defined in Q_T with respect to the norms

and

Let

be a second-order partial differential operator, where , and A_ij, B_i, C are s × s matrices of bounded functions with complex values from is the transposed conjugate matrix of A_ji.

We assume that the operator L is uniformly strong elliptic, that is, there exists a constant C > 0 such that

for all ξ ∈ ℝⁿ, η ∈ ℂ^s and a.e. (x, t) ∈ Q_T.

In this article, we study the following problem:

where f(x, t) is given.

Let us introduce the following bilinear form

Then the following Green's formula

is valid for all and a.e. t ∈ [0, T).

Definition 1.1. A function is called a generalized solution of problem (2) -(4) if and only if u|_{t = 0} = 0 and the equality

holds for all .

From (1) it follows that there exist constants µ₀ > 0, λ₀ ≥ 0 such that

holds for all and t ∈ [0, T). By substituting into (2), we can assume for convenience that λ₀ in (6) is zero. Hence, throughout the present paper we also suppose that B(., .; t) satisfies the following inequality

for all and t ∈ [0, T).

Now, let us present the main results of this article. Firstly, we give a theorem on well posedness of the problem:

Theorem 1.1. Let f ∈ L₂(Q_T, γ₀), γ₀ > 0, and suppose that the coefficients of the operator L satisfy

Then for each γ > γ₀, problem (2) -(4) has a unique generalized solution u in the space and the following estimate holds

where C is a constant independent of u and f.

Write , , . Next, we give results on the smoothness of the solution:

Theorem 1.2. Let m ∈ ℕ*, , σ = γ - γ₀, γ_k = (2k + 1)γ₀. Assume that the coefficients of L satisfy

Furthermore,

Then there exists η > 0 such that u belongs to for any |a| < η, and

where C is a constant independent of u and f.

Firstly, we will prove the existence by Galerkin's approximating method. Let be an orthogonal basis of which is orthonormal in L₂(Ω). Put

where , is the solution of the following ordinary differential system:

with the initial conditions

Let us multiply (10) by , then take the sum with respect to k from 1 to N to arrive at

Now adding this equality to its complex conjugate, we get

Utilizing (7), we obtain

By the Cauchy inequality, for an arbitrary positive number ε, we have

where C = C(ε) is a constant independent of u^N, f and t. Combining the estimates above, we get from (12) that

for a.e. t ∈ [0, T). Now write

Then (13) implies

Thus the differential form of Gronwall-Belmann's inequality yields the estimate

We obtain from (14) the following estimate:

Now multiplying both sides of this inequality by e^-γt, γ > γ₀ + ε, then integrating them with respect to t from 0 to T, we obtain

Multiplying both sides of (13) by e^-γt, then integrating them with respect to t from 0 to τ, τ ∈ (0, T), we obtain

Notice that

We employ the inequalities above to find

Since the right-hand side of (16) is independent of τ, we get

where C is a constant independent of u, f and N.

Fix any , with and write v = v¹ + v², where and (v², ω_k) = 0, k = 1, ..., N, . We have . Utilizing (10), we get

From , we can see that

Consequently,

Since this inequality holds for all , the following inequality will be inferred

Multiplying (18) by e^-γt, γ > γ₀ + ε, then integrating them with respect to t from 0 to T, and by using (17), we obtain

Combining (17) and (19), we arrive at

where C is a constant independent of f and N.

From the inequality (20), by standard weakly convergent arguments, we can conclude that the sequence possesses a subsequence weakly converging to a function , which is a generalized solution of problem (2) -(4). Moreover, it follows from (20) that estimate (8) holds.

Finally, we will prove the uniqueness of the generalized solution. It suffices to check that problem (2)-(4) has only one generalized solution u ≡ 0 if f ≡ 0. By setting v = u(., t) in identity (5) (for f ≡ 0) and adding it to its complex conjugate, we get

From (7), we have

Since u|_{t = 0} = 0, it follows from this inequality that u ≡ 0 on Q_T. The proof is complete.

Firstly, we establish the results on the smoothness of the solution with respect to time variable of the solution which claims that the smoothness depends on the smoothness of the coefficients and the right-hand side of the systems.

To simplify notation, we write

Proposition 3.1. Let h ∈ ℕ*. Assume that there exists a positive constant µ such that

(i) ,

(ii) .

Then for an arbitrary real number γ satisfying γ > γ₀, the generalized solution of problem (2)-(4) has derivatives with respect to t up to order h with , and the estimate

holds, where C is a constant independent of u and f.

Proof. From the assumptions on the coefficients of operator L and the function f, it implies that the solution of problem (10)-(11) has derivatives with respect to t up to order h + 1. We will prove by induction that

and

Firstly, we differentiate h times both sides of (10) with respect to t to find the following equality:

From the equalities above together with the initial condition (11) and assumption (ii), we can show by induction on h that

Equality (24) is multiplied by and sum k = 1, ..., N, so as to discover

Adding this equality to its complex conjugate, we get

Next, we show that inequalities (22) and (23) hold for h = 0. According to (26) (with h = 0), we have

Then the equality is rewritten in the form:

Integrating both sides of this equality with respect to t from 0 to τ, τ ∈ (0, T), employing Garding inequality (7) and Cauchy inequality, and by simple calculations, we deduce that

Thus Gronwall-Belmann's inequality yields the estimate

where . Multiplying both sides of (27) by , then integrating them with respect to t from 0 to T, we arrive at

From inequalities (27) and (28), it is obvious that (22) and (23) hold for h = 0.

Assume that inequalities (22) and (23) are valid for k = h - 1, we need to prove that they are true for k = h. With regard to equality (26), the second term in left-hand side of (26) is written in the following form:

Hence, from (26) we have

Integrating both sides of (29) with respect to t from 0 to τ, 0 < τ < T, and using the integration by parts, we find

For convenience, we abbreviate by I, II, III, IV, V the terms from the first to the fifth, respectively, of the right-hand side of (30). By using assumption (i) and the Cauchy inequality, we obtain the following estimates:

Employing the estimates above, we get from (30) that

By using (7) again, we obtain from (31) the estimate

From (32) and the induction assumptions, we get

where ε > 0 is chosen such that

By the Gronwall-Bellmann inequality, we receive from (33) that

(γ_h > γ_j, for j = 0, ..., h - 1). Now multiplying both sides of this inequality by , then integrating them with respect to τ from 0 to T, we arrive at

It means that the estimates (22) and (23) hold for k = h.

By the similar arguments in the proof of Theorem 1.1, we obtain the estimate

Then the combination between (34) and (35) produces the following inequality:

Accordingly, by again standard weakly convergent arguments, we can conclude that the sequence possesses a subsequence weakly converging to a function . Moreover, u^(k) is the k^th generalized derivative in t of the generalized solution u of problem (2)-(4). Estimate (21) follows from (36) by passing the weak convergences.   □

Next, by changing problem (2) -(4) into the Dirichlet problem for second order elliptic depending on time parameter, we can apply the results for this problem in polyhedral domains (cf. [4,5]) and our previous ones to deal with the regularity with respect to both of time and spatial variables of the solution.

Proposition 3.2. Let the assumptions of Theorem 3.1 be satisfied for a given positive integer h. Then there exists η > 0 such that belongs to for any |a| < η, k = 0, ..., h and

where C is a constant independent of u and f.

Proof. We prove the assertion of the theorem by an induction on h. First, we consider the case h = 0. Equalities (2), (3) can be rewritten in the form:

Since u satisfies

it is clear that for a.e. t ∈ (0, T), u is the solution of the Dirichlet problem for system (38) with the right-hand side for all a ≤ 1. From Theorem 4.2 in [5] (or Theorem 1.1. in [4]), it implies that there exists η > 0 such that for any |a| ≤ η. Furthermore, we have

where C is a constant independent of u, f and t. Now multiplying both sides of (40) with , then integrating with respect to t from 0 to T and using estimates from Theorem 3.1, we obtain

where C is a constant independent of u, f. Thus, the theorem is valid for h = 0. Suppose that the theorem is true for h - 1; we will prove that this also holds for h. By differentiating h times both sides of (38)-(39) with respect to t, we get

where

By the induction assumption, it implies that

and

Moreover,

by Theorem 3.1. Hence, for a.e. t ∈ (0, T), we have and the estimate

Applying Theorem 4.2 in [5] again, we conclude from (41)-(42) that and

From the inequality above and (43), it follows that

Multiplying both sides of (44) with , then integrating with respect to t from 0 to T and using Theorem 3.1 with a note that γ_k < γ_h for k = 0, 1, ..., h - 1, we obtain

where C is the constant independent of u and f. The proof is completed.   □

Proof of Theorem 1.2. We will prove the theorem by an induction on m. It is easy to see that

Hence, Proposition 3.2 implies that the theorem is valid for m = 0. Assume that the theorem is true for m - 1, we will prove that it also holds for m. It is only needed to show that

Suppose that (45) is true for s = m, m - 1, ..., j + 1, return one more to (41) (h=j), and set , we obtain

where . By the inductive assumption with respect to s, we see that

and

Thus, the right-hand side of (46) belongs to . Applying Theorem 4.2 in [5] again, we get that for a.e. t ∈ (0, T). It means that belongs to .

Furthermore, we have

Therefore,

It implies that (45) holds for s = j. The proof is complete for j = 0.

An example. In order to illustrate the results above, we show an example for the case L = -Δ, and Ω is a curvilinear polygonal domain in the plane.

Denote by A₁, A₂, ..., A_k the vertexes of Ω. Let α_j be the opening of the angle at the vertex A_j. Set

as the angle at vertex A_j with sides . Here r, θ are the polar coordinates of the point x = (x₁, x₂), noting that r(x) = ρ(x) is the distance from a point x ∈ K_j ∩ U to the set {A₁, A₂, .... A_k}, where U is a small neighbourhood of A_j.

Let be the eigenvalue of the pencil (cf. [7]) arises from the Dirichlet problem for Laplace operator via the Mellin transformation r → λ. Let η = min{η_j}. We consider the Cauchy-Dirichlet problem for the classical heat equation

where f : Q_T → ℂ is given.

Combining Theorem 1.2 and Theorem 4.4 in [5] we receive the following theorem.

Theorem 3.1. Let Ω ⊂ ℝ² be a bounded curvilinear polygonal domain in the plane. Furthermore,

Then the generalized solution u of problem (48)-(50) belongs to for any |a| < η := min η_j, as above, and u satisfies the following estimate

where C is a constant independent of u and f.

Remark: Let us notice that , , the weighed Sobolev space is defined in [[7], p. 191]. Applying Theorem 6.1.4 in [[7], p. 205] with l₂ = 2, β₂ = 1 - a, l₁ = 1, β₁ = 0, n = 2 and the strip 0 < Reλ < a < η does not contain any eigenvalue of , we obtain . It is easy to see that . Hence, the regularity of the solution of problem (48)-(50) is better than the regularity result, which can obtain from helps of Theorem 6.1.4 in [[7], p. 205].

The authors declare that they have no competing interests.

All authors typed, read, and approved the final manuscript.

This study was supported by the Vietnam's National Foundation for Science and Technology Development (NAFOSTED: 101.01.58.09).

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