Abstract
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 secondorder parabolic systems in cylinders with polyhedral base.
1 Introduction
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 secondorder 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 ℝ^{n }(n = 2, 3), and 0 < T ≤ ∞. Set Q_{T }= Ω × (0, T), S_{T }= ∂Ω × (0, T). For a vectorvalued function u = (u_{1}, u_{2}, ..., u_{s}) and p = (p_{1}, p_{2}, ..., p_{n}) ∈ ℕ^{n }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 L_{2}(Ω).
We denote by H^{m,k}(Q_{T}, γ) (γ ∈ ℝ) the weighed Sobolev space of vectorvalued 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 vectorvalued functions u ∈ C^{∞}(Q_{T}) which vanish near S_{T}.
Let ∂_{sing}Ω 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 ∂_{sing}Ω. 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 vectorvalued functions defined in Q_{T }with respect to the norms
and
Let
be a secondorder 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 ∈ ℝ^{n}, 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
From (1) it follows that there exist constants µ_{0 }> 0, λ_{0 }≥ 0 such that
holds for all and t ∈ [0, T). By substituting into (2), we can assume for convenience that λ_{0 }in (6) is zero. Hence, throughout the present paper we also suppose that B(., .; t) satisfies the following inequality
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 ∈ L_{2}(Q_{T}, γ_{0}), γ_{0 }> 0, and suppose that the coefficients of the operator L satisfy
Then for each γ > γ_{0}, 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  γ_{0}, γ_{k }= (2k + 1)γ_{0}. 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.
2 The proof of Theorem 1.1
Firstly, we will prove the existence by Galerkin's approximating method. Let be an orthogonal basis of which is orthonormal in L_{2}(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 GronwallBelmann's inequality yields the estimate
We obtain from (14) the following estimate:
Now multiplying both sides of this inequality by e^{γt}, γ > γ_{0 }+ 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 righthand 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 = v^{1 }+ v^{2}, where and (v^{2}, ω_{k}) = 0, k = 1, ..., N, . We have . Utilizing (10), we get
Consequently,
Since this inequality holds for all , the following inequality will be inferred
Multiplying (18) by e^{γt}, γ > γ_{0 }+ 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.
3 The proof of Theorem 1.2
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 righthand 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
Then for an arbitrary real number γ satisfying γ > γ_{0}, 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 GronwallBelmann'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 lefthand 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 righthand 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 GronwallBellmann 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 righthand 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 righthand 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 A_{1}, A_{2}, ..., 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 = (x_{1}, x_{2}), noting that r(x) = ρ(x) is the distance from a point x ∈ K_{j }∩ U to the set {A_{1}, A_{2}, .... 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 CauchyDirichlet 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 ⊂ ℝ^{2 }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 l_{2 }= 2, β_{2 }= 1  a, l_{1 }= 1, β_{1 }= 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].
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors typed, read, and approved the final manuscript.
Acknowledgements
This study was supported by the Vietnam's National Foundation for Science and Technology Development (NAFOSTED: 101.01.58.09).
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