Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 19 (2000), No. 1, 121–130

Initial-Boundary Value Problem for Some Coupled Nonlinear

Parabolic System of Partial Differential Equations Appearing in Thermodiffusion in Solid Body

J. Gawinecki, P. Kacprzyk and P. Bar-Yoseph

Abstract. We prove a theorem about existence, uniqueness and regularity of the solution to an initial-boundary value problem for a nonlinear coupled parabolic system consisting of two equations. Such a system appears in the thermodiffusion in solid body. In our proof we use an energy method, methods of Sobolev spaces, semigroup theory and the Banach fixed point theorem. Keywords: Linear and nonlinear parabolic systems of partial differential equations, initial-

boundary value problems, energy estimates, Sobolev spaces, semigroup theory, Banach fixed point theorem

AMS subject classification: 35 E 15, 35 K 15, 35 K 25, 35 K 50, 35 K 30

1. Introduction

We consider the initial-boundary value problem for the nonlinear coupled parabolic system of partial differential equations

c(θ_{1}, θ_{2})∂_{t}θ_{1} − a_{αβ}^{1}(θ_{1}, θ_{2}, ∇θ_{1}, ∇θ_{2})\frac{∂^{2}θ_{1}}{∂x_{α}∂x_{β}} + d(θ_{1}, θ_{2}, ∇θ_{1}, ∇θ_{2})\frac{∂θ_{2}}{∂t} = Q_{1} (1.1)

n(θ_{1}, θ_{2})∂_{t}θ_{2} − a_{αβ}^{2}(θ_{1}, θ_{2}, ∇θ_{1}, ∇θ_{2})\frac{∂^{2}θ_{2}}{∂x_{α}∂x_{β}} + d(θ_{1}, θ_{2}, ∇θ_{1}, ∇θ_{2})\frac{∂θ_{1}}{∂t} = Q_{2} (1.2)

with initial conditions

\begin{matrix}θ_{1}(0, x) & = θ_{1}^{0}(x) \\ θ_{2}(0, x) & = θ_{2}^{0}(x) \end{matrix} \}  (1.3)

and boundary conditions (Dirichlet Type)

\begin{matrix}θ_{1}(t, ·)|_{∂Ω} & = 0 \\ θ_{2}(t, ·)|_{∂Ω} & = 0 \end{matrix} \}  (1.4)

J. Gawinecki: Inst. Math. & Oper. Res., Military Univ. Techn. Warsaw, Poland P. Kacprzyk: Inst. Math. & Oper. Res., Military Univ. Techn. Warsaw, Poland P. Bar-Yoseph: Fac. Mech. Eng., Technion, Izrael Inst. Techn., Haifa, Izrael

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

where θ_{1} = θ_{1}(t, x) and θ_{2} = θ_{2}(t, x) are unknown scalar functions denoting the temperature and the chemical potential of the body, respectively, both depending on t ∈ ℝ_{+} and x ∈ Ω, Ω ⊂ ℝ^{3} being a bounded domain with smooth boundary ∂Ω, ∇θ_{1} = (∂_{1}θ_{1}, ∂_{2}θ_{1}, ∂_{3}θ_{1}) and ∇θ_{2} = (∂_{1}θ_{2}, ∂_{2}θ_{2}, ∂_{3}θ_{2}) are the gradients of the functions θ_{1} and θ_{2}, respectively, with ∂_{j} = \frac{∂}{∂x_{j}} (j = 1, 2, 3) (analogously, ∂_{t} = \frac{∂}{∂t}). Further, Q_{1} = Q_{1}(t, x) and Q_{2} = Q_{2}(t, x) denote scalar functions depending on t ∈ ℝ_{+} and x ∈ Ω, which describe the source intensities of the heat and of the diffusing mass, respectively. At last, c and n are nonlinear coefficients depending on the unknown functions θ_{1} and θ_{2}, d and a_{αβ}^{1}, a_{αβ}^{2} are nonlinear coefficients depending additionally on the gradients ∇θ_{1} and ∇θ_{2}.

For 0 < m < ∞ we denote by H^{m}(Ω) and H_{0}^{m}(Ω) the usual Sobolev spaces with norm ∥ · ∥_{m} [1]. For 1 ≤ p ≤ ∞ we denote by L^{p}(Ω) the Lebesgue function space on Ω with norm ∥ · ∥_{L^{p}}; the norm and inner product in L^{2}(Ω) are denoted by ∥ · ∥ and (·, ·), respectively.

We shall use the notation ∂_{x}^{α} = ∂_{1}^{α_{1}}∂_{2}^{α_{2}}∂_{3}^{α_{3}} (|α| = α_{1} + α_{2} + α_{3}) and denote for any integer N ≥ 0

D^{N}u = (∂_{t}^{j}∂_{x}^{α}u : j + α = N)

D̄^{N}u = (∂_{t}^{j}∂_{x}^{α}u : j + α ≤ N)
D_{x}^{N}u = (∂_{x}^{α}u : |α| = N)
D̄_{x}^{N}u = (∂_{x}^{α}u : |α| < N).

The inclusion f ∈ X for a space X with norm ∥ · ∥_{X} means that each component f_{1}, …, f_{n} of f is in X and ∥f∥_{X} = ∥f_{1}∥_{X} + … + ∥f_{n}∥_{X}. For any 0 ≤ m < ∞ and T > 0 we also use the notation |u|_{m,T} = sup_{0≤t≤T} ∥u(t)∥_{m} where ∥ · ∥_{0} denotes ∥ · ∥.

The aim of our paper is to prove existence and uniqueness (local in time) of the solution to the initial-boundary value problem (1.1) - (1.4) using methods of Sobolev spaces (cf. [6 - 8, 11, 12]). In Section 2 we present the related main theorem. In Section 3 we present a theorem about existence, uniqueness and regularity of the solution to the linearized problem associated with problem (1.1) - (1.4). Section 4 is devoted to the proof of an energy estimate for that linearized system. Finally, in Section 5 we prove the main theorem using the Banach fixed point theorem.

2. The main theorem

In this section we formulate the theorem about existence and uniqueness (local in time) of the solution to initial-boundary value problems to the nonlinear system (1.1) - (1.2). Before starting to formulate the main theorem we notice that under the condition

cn − d^{2} > 0  (2.1)

we can convert system (1.1) - (1.2) into the form

∂_{t}θ_{1} − \overline{a}_{αβ}^{11}(θ_{1}, θ_{2}, ∇θ_{1}, ∇θ_{2})\frac{∂^{2}θ_{1}}{∂x_{α}∂x_{β}} − \overline{a}_{αβ}^{12}(θ_{1}, θ_{2}, ∇θ_{1}, ∇θ_{2})\frac{∂^{2}θ_{2}}{∂x_{α}∂x_{β}}

= \overline{g}_{1}(θ_{1}, θ_{2}, ∇θ_{1}, ∇θ_{2})  (2.2)

∂_{t}θ_{2} − \overline{a}_{αβ}^{21}(θ_{1}, θ_{2}, ∇θ_{1}, ∇θ_{2})\frac{∂^{2}θ_{1}}{∂x_{α}∂x_{β}} − \overline{a}_{αβ}^{22}(θ_{1}, θ_{2}, ∇θ_{1}, ∇θ_{2})\frac{∂^{2}θ_{2}}{∂x_{α}∂x_{β}}

= \overline{g}_{2}(θ_{1}, θ_{2}, ∇θ_{1}, ∇θ_{2})  (2.3)

where \overline{a}_{αβ}^{11} = \frac{n}{δ}a_{αβ}^{1}, \overline{a}_{αβ}^{12} = −\frac{d}{δ}a_{αβ}^{2} and

\overline{g}_{1} = \frac{Q_{1}n−dQ_{2}}{δ} , \overline{a}_{αβ}^{21} = −\frac{d}{δ}a_{αβ}^{2}, \overline{a}_{αβ}^{22} = \frac{c}{δ} a_{αβ}^{2}, \overline{g}_{2} = \frac{cQ_{2}−dQ_{1}}{δ} , δ = cn − d^{2}.  (2.4)

With system (2.2) - (2.3) we associate initial conditions

\begin{matrix}θ_{1}(0, x) & = θ_{1}^{0}(x) \\ θ_{2}(0, x) & = θ_{2}^{0}(x) \end{matrix} \}  (2.5)

and boundary conditions (Dirichlet type)

\begin{matrix}θ_{1}(t, ·)|_{∂Ω} & = 0 \\ θ_{2}(t, ·)|_{∂Ω} & = 0 \end{matrix} \}  (2.6)

Now, we formulate the main theorem.

Theorem 2.1 (Local existence in time). Let the following assumptions be satisfied:

1^{∘} s ≥ [\frac{3}{2}] + 4 = 5 is an arbitrary but fixed integer.

2^{∘} ∂_{t}^{k}Q_{1}, ∂_{t}^{k}Q_{2} ∈ C^{0}([0, T], H^{s−2−k}(Ω)) for k = 0, 1, ..., s−2 and ∂_{t}^{s−1}Q_{1}, ∂_{t}^{s−1}Q_{2} ∈ L^{2}([0, T], L^{2}(Ω)).

3^{∘} There exists a constant γ > 0 such that (a_{αβ}ξ_{α}ξ_{β}η, η) ≥ γ|ξ|^{2}|η|^{2} (η ∈ ℝ^{2}, ξ ∈ ℝ^{3}) where a_{αβ} = [a_{αβ}^{ij}] (i, j = 1, 2), a_{αβ}^{ij} = a_{βα}^{ji}, with a_{αβ}^{ij}, d ∈ C^{s−1}(ℝ^{6}), c, n ∈ C^{s−1}(ℝ^{2}) and cn − d^{2} > 0.

4^{∘} θ_{1}^{0}, θ_{2}^{0} ∈ H^{s}(Ω) ∩ H_{0}^{1}(Ω), θ_{1}^{k}, θ_{2}^{k} ∈ H^{s−k}(Ω) ∩ H_{0}^{1}(Ω) (1 ≤ k ≤ s − 2) and θ_{1}^{s−1}, θ_{2}^{s−1} ∈ L^{2}(Ω) where θ_{i}^{k} = \frac{∂^{k}θ_{i}(0,·)}{∂t^{k}} (i = 1, 2) are calculated formally from system (1.1) − (1.2) and expressed with the initial data θ_{1}^{0} and θ_{2}^{0}.

Then there exists a unique solution (θ_{1}, θ_{2}) to problem (2.2)−(2.6) with the properties

\begin{matrix}θ_{i} & ∈ ∩_{k=0}^{s−2}C^{k}([0, T], H^{s−k}(Ω) ∩ H_{0}^{1}(Ω)) \\ ∂_{t}^{s−1}θ_{i} & ∈ C^{0}([0, T], L^{2}(Ω)) \\ ∂_{t}^{s−1}∇θ_{i} & ∈ L^{2}([0, T], L^{2}(Ω)) \end{matrix} \} (i = 1, 2).

The proof of Theorem 2.1 is divided into the following three steps:

Step 1^{∘}: Proof of existence, uniqueness and regularity of the solution to the initialboundary value problem for the linearized system of equations associated with system (2.2) - (2.6).

Step 2^{∘}: Proof of an energy estimate for the linearized initial-boundary value problem (2.2) - (2.6).

Step 3^{∘}: Proof of existence and uniqueness of the solution of the nonlinear initialboundary value problem (2.2) - (2.6) using the Banach fixed point theorem.

124 J. Gawinecki et al.

3. Existence, uniqueness and regularity of the solution to the linearized problem associated with problem (2.2) - (2.6)

In this section we consider the linearized problem associated with (2.2) - (2.6)

∂_{t}θ_{1} − a_{αβ}^{11}(t, x)\frac{∂^{2}θ_{1}}{∂x_{α}∂x_{β}} − a_{αβ}^{12}(t, x)\frac{∂^{2}θ_{2}}{∂x_{α}∂x_{β}} = g_{1}(t, x)  (3.1)
∂_{t}θ_{2} − a_{αβ}^{21}(t, x)\frac{∂^{2}θ_{1}}{∂x_{α}∂x_{β}} − a_{αβ}^{22}(t, x)\frac{∂^{2}θ_{2}}{∂x_{α}∂x_{β}} = g_{2}(t, x)  (3.2)

with initial conditions

\begin{matrix}θ_{1}(0, x) & = θ_{1}^{0}(x) \\ θ_{2}(0, x) & = θ_{2}^{0}(x) \end{matrix} \}  (3.3)

and boundary conditions

\begin{matrix}θ_{1}(t, ·)|_{∂Ω} & = 0 \\ θ_{2}(t, ·)|_{∂Ω} & = 0 \end{matrix} \} .  (3.4)

We consider the solvability of problem (3.1) - (3.4). At first, introducing the vector V = (θ_{1}, θ_{2})^{∗}, we can write problem (3.1) - (3.4) in the equivalent matrix form

∂_{t}V − a_{αβ}(t, x)\frac{∂^{2}V}{∂x_{α}∂x_{β}} = G(t, x)  (3.5)
V (0, x) = V^{0}(x), V (t, ·)|_{∂Ω} = 0  (3.6)

where

a_{αβ}(t, x) = ( \begin{matrix}a_{αβ}^{11}(t, x) & a_{αβ}^{12}(t, x) \\ a_{αβ}^{21}(t, x) & a_{αβ}^{22}(t, x) \end{matrix} ) (α, β = 1, 2, 3) (3.7)

and G(t, x) = (g_{1}(t, x), g_{2}(t, x))^{∗}.

Before proving an energy estimate to problem (3.1) - (3.4), we present two theorems. Theorem 3.1. Let the assumptions

\begin{matrix}\overline{D}^{1}a_{αβ}^{ij} & ∈ C^{0}([0, T] × \overline{Ω}) ∩ L^{∞}([0, T], L^{∞}(Ω)) \\ ∂_{t}∇a_{αβ}^{ij} & ∈ L^{∞}([0, T], L^{∞}(Ω)) \\ G & ∈ C^{0}([0, T], L^{2}(Ω)) \\ ∂_{t}G & ∈ L^{2}([0, T], H^{−1}(Ω)) \\ V^{0} & ∈ H_{0}^{1}(Ω) \\ V^{1} & = a_{αβ}(0)\frac{∂^{2}V^{0}}{∂x_{α}∂x_{β}} + G(0) ∈ L^{2}(Ω) \end{matrix} \}  (i, j = 1, 2)  (3.8)

and

\begin{matrix}a_{αβ}^{ij}(t, x) & = a_{βα}^{ji}(t, x) for (t, x) ∈ [0, T] × \overline{Ω} \\ (a_{αβ}ξ_{α}ξ_{β}η, η) & ≥ γ|ξ|^{2}|η|^{2} for ξ ∈ ℝ^{3}, η ∈ ℝ^{2} \end{matrix} \} (i, j = 1, 2)

be satisfied where γ > 0 is some constant. Then there exists a unique solution V = (θ_{1}, θ_{2})^{∗} to problem (3.1) − (3.4) with

\begin{matrix}θ_{1} & ∈ C^{0}([0, T], H^{2}(Ω)) ∩ H_{0}^{1}(Ω) \\ ∂_{t}θ_{1} & ∈ C^{0}([0, T], L^{2}(Ω)) \\ ∂_{t}∇θ_{1} & ∈ L^{2}([0, T], L^{2}(Ω)) \\ θ_{2} & ∈ C^{0}([0, T], H^{2}(Ω) ∩ H_{0}^{1}(Ω)) \\ ∂_{t}θ_{2} & ∈ C^{0}([0, T], L^{2}(Ω)) \\ ∂_{t}∇θ_{2} & ∈ L^{2}([0, T], L^{2}(Ω)) \end{matrix} \} .  (3.9)

Proof. It can be done by using semigroup theory and it follows directly from considerations in [3]

Now we present a higher regularity theorem connected with the solution to problem (3.1) - (3.4). The existence result is a special case of a classical theorem on local existence for parabolic systems (cf. [9]).

Theorem 3.2 (Existence, Uniqueness and Regularity). Let the following assumptions be satisfied:

1^{∘} a_{αβ}^{ij} ∈ C^{0}([0, T]×\overline{Ω})∩L^{∞}([0, T], L^{∞}(Ω)), D_{x}a_{αβ}^{ij} ∈ L^{∞}([0, T], H^{s−2}(Ω)), ∂_{t}^{k}a_{αβ}^{ij} ∈ L^{∞}([0, T], H^{s−1−k}(Ω)) (1 ≤ k ≤ s − 2) and ∂_{t}^{s−1}a_{αβ}^{ij} ∈ L^{2}([0, T], L^{2}(Ω)).

2^{∘} For θ_{1}, θ_{2} ∈ H_{0}^{1}(Ω) and all t ∈ [0, T] the inequality ∥θ_{1}∥_{1}^{2} + ∥θ_{2}∥_{1}^{2} ≤ γ_{2}\{(a_{αβ}^{ij} \frac{∂θ_{i}}{∂x_{α}} , \frac{∂θ_{j}}{∂x_{β}}) + ∥θ_{1}∥^{2} + ∥θ_{2}∥^{2}\} is satisfied for a constant γ > 0.

3^{∘} For t ∈ [0, T], −a_{αβ}^{ij}(t)\frac{∂^{2}θ_{j}}{∂x_{α}∂x_{β}} ∈ H^{k}(Ω) with θ_{1}, θ_{2} ∈ H_{0}^{1}(Ω) implis that θ_{1}, θ_{2} ∈

H^{s+2}(Ω) and ∥V ∥_{k+2} ≤ γ_{3}(∥−a_{αβ}^{ij}(t)\frac{∂^{2}V_{j}}{∂x_{α}∂x_{β}} ∥_{k} +∥V ∥) where V = (θ_{1}, θ_{2}), 0 ≤ k ≤ s−2 and γ_{3} > 0 is some constant.

4^{∘} ∂_{t}^{k}g_{i} ∈ C^{0}([0, T], H^{s−2−k}(Ω)) (0 ≤ k ≤ s−2) and ∂_{t}^{s−1}g_{i} ∈ L^{2}([0, T], H^{−1}(Ω)) (i = 1, 2), where s ≥ [\frac{3}{2}] + 4 = 5 is an arbitrary but fixed fixed integer.

Then there exists a unique solution V = (θ_{1}, θ_{2})^{∗} to the initial-boundary value problem (3.1) − (3.4) with the properties

\begin{matrix}∂_{t}^{k}θ_{i} & ∈ C^{0}([0, T], H^{s−2−k}(Ω) ∩ H_{0}^{1}(Ω)) (0 ≤ k ≤ s − 2) \\ ∂_{t}^{s−1}θ_{i} & ∈ C^{0}([0, T], L^{2}(Ω)) \\ ∂_{t}^{s−1}∇θ_{i} & ∈ L^{2}([0, T], L^{2}(Ω)) \end{matrix} \}  (i = 1, 2). (3.10)

Proof. It is based on Theorem 2.1, the assumption of Theorem 2.2 and mathematical induction

Remark 3.1. In order to obtain the solution of problem (3.1) - (3.4) with regularity (3.10) the initial data must satisfy the compatibility conditions

V^{k} = (θ_{1}^{k}, θ_{2}^{k}) ∈ (H^{s−k}(Ω) ∩ H_{0}^{1}(Ω)) × (H^{s−k}(Ω) ∩ H_{0}^{1}(Ω))

where k = 0, 1, ..., s − 2 and

V^{s−1} = (θ_{1}^{s−1}, θ_{2}^{s−1}) ∈ L^{2}(Ω) × L^{2}(Ω).  (3.11)

We define V^{k} successively by

V^{k} = ∑_{j=0}^{k−1} ({k − 1 \atop j})∂_{k}^{j}a_{αβ}(0)\frac{∂^{2}V^{k−1−j}}{∂x_{α}∂x_{β}} + ∂_{t}^{k−1}G(0) (k ≥ 1).

4. An energy estimate for problem (3.1) - (3.4)

We start with the formulation of the following

Theorem 4.1 (Energy estimate). Let the conditions of Theorem 2.2 be fulfilled. Then the solution V = (θ_{1}, θ_{2}) to the initial-boundary value problem (3.1) − (3.4) established in Theorem 3.2 satisfies the inequality

\begin{matrix}∑_{k=0}^{s−2} |∂_{t}^{k}θ_{1}|_{s−k,T}^{2} \\ + ∑_{k=0}^{s−2} |∂_{t}^{k}θ_{2}|_{s−k,T}^{2} + |∂_{t}^{s−1}θ_{1}|_{0,T}^{2} + |∂_{t}^{s−1}θ_{2}|_{0,T}^{2} \\ + ∫_{0}^{t} [∥∂_{t}^{s−1}∇θ_{1}(τ)∥^{2} + ∥∂_{t}^{s−1}∇θ_{2}(τ)∥^{2}]dτ ≤ K_{3}M_{0}e^{K_{4}η(T)} \end{matrix}  (4.1)

where

\begin{aligned}M_{0} & = (1 + T)\{ ∑_{k=0}^{s−2}(∥θ_{1}^{k}∥_{s−k}^{2} + ∥θ_{2}^{k}∥_{s−k}^{2}) + ∥θ_{1}^{s−1}∥^{2} \\ & + ∥θ_{2}^{s−1}∥^{2} + ∥\overline{D}^{s−2}g_{1}∥_{0,T}^{2} + ∥\overline{D}^{s−2}g_{2}∥_{0,T}^{2} \\ & + ∫_{0}^{T} [∥∂_{t}^{s−1}g_{1}(τ)∥_{H^{−1}}^{2} + ∥∂_{t}^{s−1}g_{2}(τ)∥_{H^{−1}}^{2}]dτ\} \end{aligned}  (4.2)

and K_{3} = K_{3}(P_{0}, γ_{2}, γ_{3}), K_{4} = K_{4}(P, γ_{2}, γ_{3}) are positive constants depending continuously on P_{0}, P, γ_{2}, γ_{3} are constants defined in the assumption of Theorem 3.1,

P = sup_{0≤t≤T} ∑_{i,j=1}^{3} ∥a_{αβ}^{ij}(t)∥_{L^{∞}} + ∑_{i,j=1}^{2} ∥D_{x}a_{αβ}^{ij}∥_{s−2,T}

+ ∑_{k=1}^{s−2} ∑_{i,j=1}^{2} |∂_{t}^{k}a_{αβ}^{ij}|_{s−1−k} + ∫_{0}^{T} ∑_{i,j=1}^{2} |∂_{t}^{s−1}a_{αβ}^{ij}(τ)|^{2}dτ

P_{0} = ∑_{i,j=1}^{2} ∥a_{αβ}^{ij}(0)∥_{L^{∞}} + ∑_{i,j=1}^{2} ∥D_{x}a_{αβ}^{ij}(0)∥_{s−3}

and

η(T) = T (1 + T).  (4.3)

Proof. It can be found in [7]

5. Proof of Theorem 2.1

The proof of Theorem 2.1 is based on the Banach fixed point theorem. At first, we define Z(N, T) as the set of functions (θ_{1}, θ_{2}) which satisfy

\begin{matrix}∂_{t}^{k}θ_{i} & ∈ L^{∞}([0, T], H^{s−k}(Ω)) (0 ≤ k ≤ s − 2) \\ ∂_{t}^{s−1}θ_{i} & ∈ L^{∞}([0, T], L^{2}(Ω)) \\ ∂_{t}^{s−1}∇θ_{i} & ∈ L^{2}([0, T], L^{2}(Ω)) \end{matrix} \}  (i = 1, 2)  (5.1)

(s ≥ [\frac{3}{2}] + 4 = 5) with boundary and initial conditions of the form

\begin{matrix}θ_{i}|_{∂Ω} & = 0 \\ ∂_{t}^{k}θ_{i}(0, x) & = θ_{i}^{k}(x) \end{matrix} \} (i = 1, 2; 0 ≤ k ≤ s − 2)

and the inequality

\begin{matrix}∑_{l=0}^{s−2} |∂_{t}^{k}θ_{1}|_{s−k,T}^{2} + |∂_{t}^{s−2}θ_{1}|_{0,T}^{2} \\ + ∑_{l=0}^{s−2} |∂_{t}^{k}θ_{2}|_{s−k,T}^{2} + |∂_{t}^{s−2}θ_{2}|_{0,T}^{2} \\ + ∫_{0}^{T} [∥∂_{t}^{s−1}∇θ_{1}(τ)∥^{2} + ∥∂_{t}^{s−1}∇θ_{2}(τ)∥^{2}]dτ ≤ N \end{matrix}  (5.2)

for N large enough. Now, we consider the system of equations

∂_{t}θ_{1} − a_{αβ}^{11} \frac{∂^{2}θ_{1}}{∂x_{α}∂x_{β}} − a_{αβ}^{12} \frac{∂^{2}θ_{2}}{∂x_{α}∂x_{β}} = g_{1}  (5.3)
∂_{t}θ_{2} − a_{αβ}^{21} \frac{∂^{2}θ_{1}}{∂x_{α}∂x_{β}} − a_{αβ}^{22} \frac{∂^{2}θ_{2}}{∂x_{α}∂x_{β}} = g_{2}  (5.4)

with initial and boundary conditions (2.5) and (2.6) where

\begin{matrix}a_{αβ}^{11} := \overline{a}_{αβ}^{11}(\overline{θ}_{1}, \overline{θ}_{2}, ∇\overline{θ}_{1}, ∇\overline{θ}_{2})  a_{αβ}^{12} := \overline{a}_{αβ}^{12}(\overline{θ}_{1}, \overline{θ}_{2}, ∇\overline{θ}_{1}, ∇\overline{θ}_{2}) \\ a_{αβ}^{21} := \overline{a}_{αβ}^{21}(\overline{θ}_{1}, \overline{θ}_{2}, ∇\overline{θ}_{1}, ∇\overline{θ}_{2})  a_{αβ}^{22} := \overline{a}_{αβ}^{22}(\overline{θ}_{1}, \overline{θ}_{2}, ∇\overline{θ}_{1}, ∇\overline{θ}_{2}) \end{matrix} \}

and

\begin{matrix}g_{1} & := \overline{g}_{1}(\overline{θ}_{1}, \overline{θ}_{2}, ∇\overline{θ}_{1}, ∇\overline{θ}_{2}, t, x) \\ g_{2} & := \overline{g}_{2}(\overline{θ}_{1}, \overline{θ}_{2}, ∇\overline{θ}_{1}, ∇\overline{θ}_{2}, t, x) \end{matrix} \} .  (5.5)

Applying Theorem 3.2 to problem (5.3) - (5.5), (2.3) - (2.4) we can see that there exists a mapping σ such that

σ : Z(N, T) ∋ (\overline{θ}_{1}, \overline{θ}_{2}) → σ(\overline{θ}_{1}, \overline{θ}_{2}) = (θ_{1}, θ_{2}).

Next we prove that σ maps Z(N, T) into itself under the conditions that N is large and T small enough. For this we introduce the notation

\begin{aligned}E_{0} & = ∑_{k=0}^{s−2} ∥θ_{1}^{k}∥_{s−k}^{2} + ∥θ_{1}^{s−1}∥^{2} + ∑_{k=0}^{s−2} ∥θ_{2}^{k}∥_{s−k}^{2} + ∥θ_{2}^{s−1}∥^{2} \\ & + ∑_{k=0}^{s−2} |∂_{t}^{k}(θ_{1}, θ_{2})|_{s−2−k,T}^{2} + ∑_{k=0}^{s−2} |∂_{t}^{k}(Q_{1}, Q_{2})|_{s−2−k,T}^{2} \\ & + ∫_{0}^{T} ∥∂_{t}^{s−1}(Q_{1}, Q_{2})∥^{2}dτ. \end{aligned}  (5.6)

After some calculations and taking into account inequality N(t) = N(0) + ∫_{0}^{t} ∂_{τ} N(τ) dτ we get

\begin{matrix}∑_{k=0}^{s−2} |∂_{t}^{k}\overline{g}_{1}|_{s−2−k,T} + ∑_{k=0}^{s−2} |∂_{t}^{k}\overline{g}_{2}|_{s−2−k,T} + ∫_{0}^{T}(∥∂_{t}^{s−1}\overline{g}_{1}∥_{s−1}^{2} + ∥∂_{t}^{s−1}\overline{g}_{2}∥_{s−1}^{2})dt \\ ≤ C(E_{0}) + C(N)T (1 + T). \end{matrix} (5.7)

Taking into account that

K_{3}, K_{4} ≤ C(E_{0}) + C(N)T (1 + T)  (5.8)

and putting (5.6) and (5.7) into the energy estimate, we obtain

\begin{matrix}∑_{k=0}^{s−2} |∂_{t}^{k}θ_{1}|_{s−k,T} + ∑_{k=0}^{s−2} |∂_{t}^{k}θ_{2}|_{s−k,T} + |∂_{t}^{s−1}θ_{1}|_{0,T} + |∂_{t}^{s−1}θ_{2}|_{0,T} \\ + ∫_{0}^{T} (∥∂_{τ}^{s−1}∇θ_{1}∥^{2} + ∥∂_{τ}^{s−1}∇θ_{2}∥)dτ \\ ≤ K(E_{0}, γ_{2}, γ_{3})(1 + C(N)T (1 + T))e^{C(N)T (1+T ^{\frac{1}{2}} +T^{4}+T ^{\frac{3}{2}} )}. \end{matrix}  (5.9)

Now we choose N such that K(E_{0}, γ_{2}, γ_{3}) ≤ \frac{N}{2} . Then we can notice that

α(T) = (1 + C(N)T (1 + T)^{2})e^{C(N)T ^{\frac{1}{2}} (1+T ^{\frac{1}{2}} +T +T ^{\frac{3}{2}} )} < 2

and for T small enough (α(0) = 1) we conclude that

σ(Z(N, T)) ⊂ Z(N, T).  (5.10)

Now we prove that

σ : Z(N, T) → Z(N, T)  (5.11)

is even a contraction mapping. For this we define the matric space (complete) (W, ρ) where

W = {(θ_{1}, θ_{2}) : θ_{1}, θ_{2} ∈ L^{∞}([0, T], L^{2}(Ω)), ∇θ_{1}, ∇θ_{2} ∈ L^{2}([0, T], L^{2}(Ω))} (5.12)

Initial-Boundary Value Problem 129

and

\begin{aligned}ρ((\overline{θ}_{1}, \overline{θ}_{2}), (θ_{1}, θ_{2})) & = |\overline{θ}_{1} − θ_{1}|_{0,T} + |\overline{θ}_{2} − θ_{2}|_{0,T} \\ & + ∫_{0}^{T}∥∇(\overline{θ}_{1} − θ_{1})(τ)∥^{2}dτ + ∫_{0}^{T}∥∇(\overline{θ}_{2} − θ_{2})(τ)∥^{2}dτ. \end{aligned} (5.13)

The set Z(N, T) is a closed subset in (W, ρ). Let (\overline{θ}_{1}, \overline{θ}_{2}), (\overline{θ}_{1}^{∗}, \overline{θ}_{2}^{∗}) ∈ Z(N, T) and let

\begin{matrix}σ(\overline{θ}_{1}, \overline{θ}_{2}) & = (θ_{1}, θ_{2}) ∈ Z(N, T) \\ σ(\overline{θ}_{1}^{∗}, \overline{θ}_{2}^{∗}) & = (θ_{1}^{∗}, θ_{2}^{∗}) ∈ Z(N, T) \end{matrix} \} .  (5.14)

Subtracting by side the corespoding system for θ_{1}, θ_{2} and θ_{1}^{∗}, θ_{2}^{∗} we get

\begin{matrix}∂_{t}(θ_{i} − θ_{i}^{∗}) − a_{αβ}^{ij}(\overline{θ}_{1}, \overline{θ}_{2}, ∇\overline{θ}_{1}, ∇\overline{θ}_{2})\frac{∂^{2}(θ_{j} − θ_{j}^{∗})}{∂x_{α}∂x_{β}} \\ = (a_{αβ}^{ij}(\overline{θ}_{1}^{∗}, \overline{θ}_{2}^{∗}, ∇\overline{θ}_{1}^{∗}, ∇\overline{θ}_{2}^{∗}) − \overline{a}_{αβ}^{ij}(\overline{θ}_{1}, \overline{θ}_{2}, ∇\overline{θ}_{1}, ∇\overline{θ}_{2})) · \frac{∂^{2}θ_{j}^{∗}}{∂x_{α}∂x_{β}} \\ + \overline{g}_{i}(\overline{θ}_{1}, \overline{θ}_{2}, ∇\overline{θ}_{1}, ∇\overline{θ}_{2})(x, t) − g_{i}(\overline{θ}_{1}^{∗}, \overline{θ}_{2}^{∗}, ∇\overline{θ}_{1}^{∗}, ∇\overline{θ}_{2}^{∗})(x, t) \end{matrix}  (5.15)

for i = 1, 2. Using the fact that

sup_{0≤t≤T} ∥\overline{D}^{2}(\overline{θ}_{1}, \overline{θ}_{2}, \overline{θ}_{1}^{∗}, \overline{θ}_{2}^{∗}, θ_{1}, θ_{2}, θ_{1}^{∗}θ_{2}^{∗})∥ ≤ CN and \begin{matrix}(θ_{i} − θ_{i}^{∗})|_{∂Ω} & = 0 \\ (θ_{i} − θ_{i}^{∗})(0, x) & = 0 \end{matrix} \}

and taking into account the mean value theorem

\begin{aligned}C(θ_{1}, θ_{2}) − C(θ_{1}^{∗}, θ_{2}^{∗}) & = C(θ_{1}^{∗} + (θ_{1} − θ_{1}^{∗}), θ_{2}^{∗} + (θ_{2} − θ_{2}^{∗})) − C(θ_{1}^{∗}, θ_{2}^{∗}) \\ & = ∇_{ξ}C(ξ) · (θ − θ^{∗}), \end{aligned}

after multiplying equation (5.13) by θ_{i} − θ_{i}^{∗} and intergrating on [0, t] × Ω we get

\begin{matrix}∥θ_{1} − θ_{1}^{∗}∥^{2} + ∫_{0}^{t} ∥∇(θ_{1} − θ_{1}^{∗})∥^{2}dτ + ∥θ_{2} − θ_{2}^{∗}∥^{2} + ∫_{0}^{t} ∥∇(θ_{2} − θ_{2}^{∗})∥^{2}dτ \\ ≤ C(N)(1 + \frac{1}{T^{1/2}} ) ∫_{0}^{t} (∥θ_{1} − θ_{1}^{∗}∥^{2} + ∥θ_{2} − θ_{2}^{∗}∥^{2})dτ \\ + (T^{\frac{1}{2}} (1 + T)[|\overline{θ}_{1} − \overline{θ}_{1}^{∗}|_{0,T}^{2} + |\overline{θ}_{2} − \overline{θ}_{2}^{∗}|_{0,T}^{2} ] \\ + ∫_{0}^{t} (∥∇(\overline{θ}_{1} − \overline{θ}_{1}^{∗})∥^{2} + ∥∇(\overline{θ}_{2} − \overline{θ}_{2}^{∗})∥^{2})dτ \\ + (1 + \frac{1}{T^{1/2}} ) ∫_{0}^{t}∫_{0}^{s} (∥∇(θ_{1} − θ_{1}^{∗})∥^{2} + ∥∇(θ_{2} − θ_{2}^{∗})∥^{2})dτdt. \end{matrix}  (5.16)

Applying to (5.14) the Growall inquality we get

\begin{matrix}|θ_{1} − θ_{1}^{∗}|_{0,T}^{2} + |θ_{2} − θ_{2}^{∗}|_{0,T}^{2} + ∫_{0}^{T}(∥∇(θ_{1} − θ_{1}^{∗})∥^{2} + ∥∇(θ_{2} − θ_{2}^{∗})∥^{2})dτ \\ ≤ ε[|\overline{θ}_{1} − \overline{θ}_{1}^{∗}|_{0,T}^{2} + |\overline{θ}_{2} − \overline{θ}_{2}^{∗}|_{0,T}^{2} + ∫_{0}^{T}(∥∇(\overline{θ}_{1} − \overline{θ}_{1}^{∗})∥^{2} + ∥∇(\overline{θ}_{2} − \overline{θ}_{2}^{∗})∥)dτ] \end{matrix} (5.17)

where ε = C(N)T^{\frac{1}{2}} (1 + T)e^{C(N)(T +T ^{\frac{1}{2}} )}. So choosing T small enough we obtain ε < 1. So it means that the mapping σ is a contraction. This ends the proof of Theorem 2.1.

Acknowledgment. The main part of the work on this paper was done in July 1997, in scientific cooperation with Professor Phinas Bar-Yoseph during a stay of the first author at the Technion as a visiting professor.

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Received 17.07.1998; in revised form 09.11.1999