PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 91(105) (2012), 111–123 DOI: 10.2298/PIM1205111J

FORMULATION AND ANALYSIS OF A PARABOLIC TRANSMISSION PROBLEM ON DISJOINT INTERVALS

Boško S. Jovanović and Lubin G. Vulkov

Abstract. We investigate an initial-boundary-value problem for one dimensional parabolic equations in disjoint intervals. Under some natural assumptions on the input data we proved the well-posedness of the problem. Nonnegativity and energy stability of its weak solutions are also studied.

1. Introduction

It is well known that the transfer of energy or mass is fundamental for many biological, chemical, environmental and industrial processes. The basic transport mechanisms of such processes are diffusion (or dispersion) and bulkflow. Therefore, the corresponding flux has two components: a diffusive one and a convective one. Here we pay attention to diffusion in one-dimensional domain with layers. Layers with material properties which significantly differ from those of surrounding medium appear in variety of applications [1, 2, 3, 4, 7, 8, 12, 14, 17]. The layer may have structural role (as in the case of glue), a thermal role (as in the case of thin thermal insulator), an electromagnetic or optical role, etc., depending on the application. Traditionally there are two ways of handling such layers in the numerical modelling: either they are fully modelled or, they are totally ignored. We use a method proposed in [7, 8] of modelling of a thin layer as an interface. The effect of the layer on the solution is modelled by means of nonlocal jump conditions across the interface. In order to explain the method proposed in this paper for mathematical modelling of layer phenomena we consider in Section 2 a physical model. As a result the following initial-boundary-value problem (IBVP) arises: find functions u_{1}(x, t), u_{2}(x, t) that satisfy the parabolic equations

(1.1)  \frac{∂u_{1}}{∂t} − \frac{∂}{∂x}( p_{1}(x)\frac{∂u_{1}}{∂x} ) + q_{1}(x)u_{1} = f_{1}(x, t), x ∈ Ω_{1} ≡ (a, b), t > 0,

2010 Mathematics Subject Classification: 35K20; 35K99; 35D05.

Key words and phrases: transmission problem, disjoint intervals, weak solution, nonnegativity preservation.

(1.2)  \frac{∂u_{2}}{∂t} − \frac{∂}{∂x}( p_{2}(x)\frac{∂u_{2}}{∂x} ) + q_{2}(x)u_{2} = f_{2}(x, t), x ∈ Ω_{2} ≡ (c, d), t > 0,

where −∞ < a < b < c < d < +∞, and the internal boundary (or nonlocal interface jump) conditions

(1.3)  p_{1}(b)\frac{∂u_{1}(b, t)}{∂x} + α_{1}u_{1}(b, t) = β_{1}u_{2}(c, t) + γ_{1}(t),

(1.4)  − p_{2}(c)\frac{∂u_{2}(c, t)}{∂x} + α_{2}u_{2}(c, t) = β_{2}u_{1}(b, t) + γ_{2}(t).

These two conditions have the form of Robin–Dirichlet mixed boundary conditions (see the review of Givoli [7], where Robin–Dirichlet conditions have been incorporated in a finite element formulation in order to eliminate an infinite domain, a singular domain, or a substructure from computational domain). Finally, in order to complete the IBVP we pose the simplest external boundary conditions

(1.5)  u_{1}(a, t) = 0, u_{2}(d, t) = 0,

and initial conditions

(1.6)  u_{1}(x, 0) = u_{10}(x), u_{2}(x, 0) = u_{20}(x).

Throughout the paper we assume that the data satisfy the usual regularity and ellipticity conditions

p_{i}(x), q_{i}(x) ∈ L_{∞} (Ω_{i}), i = 1, 2, (1.7)

p_{i}(x) ⩾ p_{0i} > 0, a.e. in Ω_{i}, i = 1, 2, (1.8)

α_{i} > 0 , β_{i} > 0, i = 1, 2. (1.9)

In some cases we shall require that

(1.10)  q_{i}(x) ⩾ 0, a.e. in Ω_{i}, i = 1, 2,

and/or

(1.11)  β_{1}β_{2} ⩽ α_{1}α_{2}.

The aim of the present paper is to study the well-posedness of the IBVP (1.1)– (1.6) and some of its properties as nonnegativity and energy stability of the solutions. Finite difference schemes for approximation of (1.1)–(1.6) are considered in [10, 11].

An outline of the paper is as follows. In Section 2 we discuss a heat-mass transfer problem and show how in a natural way can arise problem (1.1)–(1.6). In Section 3 the well-posedness (existence, uniqueness and regularity of the solution) for problem (1.1)–(1.6) is studied. A specific spectral problem is used. Energy stability for solutions to problem (1.1)–(1.6) is also discussed. The nonnegativity of solutions to (1.1)–(1.11) is studied in Section 4.

2. A Motivated Heat-Mass Transfer Problem

We know that conductive heat transfer is nothing else but the movement of thermal energy through the corresponding medium (material) from the more energetic particles to the others [3, 4, 17]. Of course, the local temperature is given by the energy of the molecules situated at that place. So, thermal energy is transferred from points with higher temperature to other points. If the temperature of some area of the medium increases, then the random molecular motion becomes more intense in that area. Thus a transfer of thermal energy is produced, which is called heat diffusion. The corresponding conductive heat flux Q(x, t), is given by the Fourier’s rate law Q = −pu_{x} = −p\frac{∂u}{∂x} , where p is the thermal conductivity of the medium, and u = u(x, t) is the temperature at point x at time instant t. We shall assume that p depends on x only. The first law of thermodynamics (conservation of energy) gives in a standard manner the transformation of the energy from one form (e.g. mechanical, electrical, etc.) into heat. The right hand side of the above equation represents the stored heat. In other words, the heat equation looks like

ε\frac{∂u}{∂t} − \frac{∂}{∂x}( p(x)\frac{∂u}{∂x}) = f(x, t),

where ε = ρc, ρ is the density, and c is the specific heat capacity. Note that the same partial differential equation is a model of mass transfer, but in this case u(x, t) represents the mass density of the material at a point x, at time t. Instead of Fourier law, a similar law is available in this case, which is called Fick’s law of mass diffusion. An additional term q(x)u depending on the density may appear in the mass equation due to the reaction

(2.1)  ε\frac{∂u}{∂t} − \frac{∂}{∂x}( p(x)\frac{∂u}{∂x}) + q(x)u = f(x, t).

For the case of an interface joining two media with thermal conductivities p_{i} (i = 1, 2) the heat flows Q_{i} (even temperatures u_{i}) have a discontinuity jump at the interface (a point in 1D case, a curve in 2D case and a surface in the 3D case). We shall now discuss a 1D model of two media of non-stationary transfer separated by a layer where the temperature field is almost stationary, i.e., ∂u/∂t ≡ 0 or the quantity ε = ρc is very small. Then equation (2.1) in the layer takes the form:

(2.2)  − \frac{∂}{∂x}( p_{L}(x)\frac{∂u_{L}}{∂x} ) + q_{L}(x)u_{L} = f_{L}(x, t).

Let us suppose that the heat (or mass) equations of media 1, 2 are given by (1.1), (1.2) respectively, and in the intermediate layer (b, c), by (2.2). On the two ends of the layer we have the continuity conditions

u_{1}(b, t) = u_{L}(b, t), u_{L}(c, t) = u_{2}(c, t), (2.3)

p_{1}(b)\frac{∂u_{1}}{∂x} (b, t) = p_{L}(b)\frac{∂u_{L}}{∂x} (b, t), p_{L}(c)\frac{∂u_{L}}{∂x} (c, t) = p_{2}(c)\frac{∂u_{2}}{∂x} (c, t). (2.4)

Conditions (2.3) enforce the continuity of the primary variable u (e.g., temperature), whereas conditions (2.4) require the continuity of the ‘flux’ Q. In the layer (b, c) we solve equation (2.2) analytically. Its general solution is of the form

(2.5)  u_{L}(x, t) = C_{1}(t)v_{1}(x) + C_{2}(t)v_{2}(x) + w(x, t),

where v_{1}, v_{2}, w are known functions, and C_{1}(t), C_{2}(t) are unknown functions. From (2.5) we can write

[ \begin{matrix}v_{1}(b) & v_{2}(b) \\ v_{1}(c) & v_{2}(c) \end{matrix}] · [ \begin{matrix}C_{1}(t) \\ C_{2}(t) \end{matrix}] = [ \begin{matrix}u_{L}(b, t) − w(b, t) \\ u_{L}(c, t) − w(c, t) \end{matrix}] .

By solving this set of equations we express C_{1}(t) and C_{2}(t) in terms of u_{L}(b, t), u_{L}(c, t) and w(b, t), w(c, t). Then, from (2.3)–(2.5) we obtain conditions (1.3) and (1.4), where

α_{1} = \frac{p_{L}(b)Δ_{1}(c, b)}{Δ(b, c)} , β_{1} = \frac{p_{L}(b)Δ_{1}(b, b)}{Δ(b, c)},

α_{2} = \frac{p_{L}(c)Δ_{1}(b, c)}{Δ(b, c)} , β_{2} = \frac{p_{L}(c)Δ_{1}(c, c)}{Δ(b, c)} ,

γ_{1}(t) = \frac{p_{L}(b)}{Δ(b, c)} ( Δ_{1}(c, b) w(b, t) − Δ_{1}(b, b) w(c, t) + Δ(b, c) \frac{∂w}{∂x}(b, t)) ,

γ_{2}(t) = \frac{p_{L}(c)}{Δ(b, c)} ( Δ_{1}(c, c) w(b, t) − Δ_{1}(b, c) w(c, t) + Δ(b, c) \frac{∂w}{∂x}(c, t)) ,

Δ(b, c) = |\begin{matrix}v_{1}(b) & v_{2}(b) \\ v_{1}(c) & v_{2}(c) \end{matrix}| = v_{1}(b)v_{2}(c) − v_{1}(c)v_{2}(b),

Δ_{1}(r, s) = | \begin{matrix}v_{1}(r) & v_{2}(r) \\ \frac{dv_{1}}{dx} (s) & \frac{dv_{2}}{dx} (s) \end{matrix}| = v_{1}(r)\frac{dv_{2}}{dx}(s) − v_{2}(r)\frac{dv_{1}}{dx}(s).

Lemma 2.1. For the constants α_{i} and β_{i} inequalities (1.9) hold.

Proof. The functions v_{1}(x) and v_{2}(x) are two linearly independent solutions of the corresponding homogeneous ordinary differential equation

Lv ≡ −\frac{d}{dx}( p_{L}(x)\frac{dv}{dx}) + q_{L}(x)v = 0, x ∈ (b, c).

For example, we can choose v_{1}(x) and v_{2}(x) as the solutions of the Cauchy problems

Lv_{1} = 0, b < x < c, v_{1}(b) = 0, p_{L}(b)\frac{dv_{1}}{dx}(b) = 1,
Lv_{2} = 0, b < x < c, v_{2}(c) = 0, p_{L}(c)\frac{dv_{2}}{dx}(c) = −1.

Then maximum principle arguments [6] imply v_{1}(c) > 0, \frac{dv_{1}}{dx} (b) > 0, \frac{dv_{1}}{dx} (c) > 0, v_{2}(b) > 0, \frac{dv_{2}}{dx} (b) < 0 and \frac{dv_{2}}{dx} (c) < 0, wherefrom follows (1.9). □

Remark 2.1. If one formally sets β_{1} = 0 (or β_{2} = 0) problem (1.1)–(1.6) can be decoupled into two independent initial-boundary-value problems. Indeed, in that case u_{1} satisfies equation (1.1), boundary conditions of the Dirichlet and Robin type

u_{1}(a, t) = 0, p_{1}(b)\frac{∂u_{1}(b, t)}{∂x} + α_{1}u_{1}(b, t) = γ_{1}(t),

and the initial condition u_{1}(x, 0) = u_{10}(x). Such problems are already well studied (see e.g., [16, 18, 20]), where, for the well-posedness, α_{1} > 0 is assumed. When u_{1} is determined, one obtains an analogous mixed Dirichlet–Robin initial-boundary-value problem for u_{2}.

Remark 2.2. Introducing new unknown functions

U_{1}(x, t) = u_{1}(x, t) − A_{1}(t)(x − a)(x − b), U_{2}(x, t) = u_{2}(x, t) − A_{2}(t)(x − c)(x − d),

where A_{1}(t) = γ_{1}(t)/[(b − a)p_{1}(b)] and A_{2}(t) = γ_{2}(t)/[(d − c)p_{2}(c)], one obtains an analogous initial-boundary-value problem with homogeneous internal boundary conditions. In such a way, without loss of generality, we can set γ_{1}(t) = γ_{2}(t) = 0 in (1.3)–(1.4).

3. Well-Posedness of Problem (1.1)–(1.6)

After the formulation and explanation of the physical meaning of problem (1.1)–(1.6), we come up to the following theoretical questions: (a) existence, uniqueness and qualitative properties (as smoothness, positivity, energy stability, etc.) of the solution; (b) construction of analytical and numerical methods for approximate solution of the problem. This paper is concerned with the first group of questions. We begin with the well-posedness, i.e., existence and uniqueness of solution in appropriate Sobolev spaces.

3.1. An Auxiliary Spectral Problem. The problem is to find the triple [λ; v = (v_{1}, v_{2})] ∈ R × H^{1}(Ω_{1}) × H^{1}(Ω_{2}) which satisfies the differential equations

(3.1)  L_{1}v_{1} ≡ −\frac{d}{dx}( p_{1}(x)\frac{dv_{1}}{dx}) + q_{1}(x)v_{1}(x) = λv_{1}(x), x ∈ Ω_{1},
(3.2)  L_{2}v_{2} ≡ −\frac{d}{dx}( p_{2}(x)\frac{dv_{2}}{dx}) + q_{2}(x)v_{2}(x) = λv_{2}(x), x ∈ Ω_{2},

with Dirichlet’s classical boundary conditions

(3.3)  v_{1}(a) = 0, v_{2}(d) = 0,

as well as the nonlocal Robin–Dirichlet boundary conditions

(3.4)  \begin{aligned}l_{1}(v_{1}, v_{2}) ≡ p_{1}(b)\frac{dv_{1}}{dx}(b) + α_{1}v_{1}(b) − β_{1}v_{2}(c) & = 0, \\ l_{2}(v_{1}, v_{2}) ≡ −p_{1}(c)\frac{dv_{2}}{dx}(c) + α_{2}v_{2}(c) − β_{2}v_{1}(b) & = 0. \end{aligned}

Assuming that conditions (1.9) hold, consider the product space L = L_{2}(Ω_{1}) × L_{2}(Ω_{2}), endowed with the inner product and associated norm

(u, v)_{L} = ∑_{i=1}^{2} β_{3−i} (u_{i}, v_{i})_{L_{2} (Ω_{i} )} , ∥v∥_{L} = (v, v)_{L}^{1/2} ,

where

(u_{i}, v_{i})_{L_{2} (Ω_{i} )} = ∫_{Ω_{i}} u_{i}v_{i} dx, i = 1, 2.

We can identify v ∈ L with a scalar function in Ω = Ω_{1} ∪ Ω_{2}, by v : Ω → R, v|_{Ω_{i}} = v_{i}, i = 1, 2. We introduce the product space

H^{1} = {v = (v_{1}, v_{2}) | v_{i} ∈ H^{1}(Ω_{i}) and v_{1}(a) = 0, v_{2}(d) = 0}

endowed with the inner product and associated norm

(u, v)_{H^{1}} = ∑_{i=1}^{2} β_{3−i} [(u_{i}, v_{i})_{L_{2} (Ω_{i} )} + ( \frac{du_{i}}{dx} , \frac{dv_{i}}{dx})_{L_{2} (Ω_{i} )} ], ∥u∥_{H^{1}} = (u, u)_{H^{1}}^{1/2} .

We have the following assertion.

Lemma 3.1. Under the regularity conditions (1.7)–(1.9), the spectral problem (3.1)–(3.4) is formally equivalent to the following variational problem: find [λ, v] ∈ R × H^{1} such that

(3.5)  A(v, w) ≡ [v, w] + Z(v, w) = λ(v, w)_{L}, ∀w ∈ H^{1},

where

[u, v] = ∑_{i=1}^{2} β_{3−i} [u_{i}, v_{i}]_{i}, [u_{i}, v_{i}]_{i} = ∫_{Ω_{i}} ( p_{i} \frac{du_{i}}{dx} \frac{dv_{i}}{dx} + q_{i}u_{i}v_{i}) dx, i = 1, 2,

Z(v, w) = β_{2}α_{1}v_{1}(b) w_{1}(b) + β_{1}α_{2}v_{2}(c) w_{2}(c) − β_{1}β_{2}[v_{1}(b) w_{2}(c) + v_{2}(c) w_{1}(b)].

Proof. Consider the eigentriple [λ; v_{1}, v_{2}] of (3.1)–(3.4). Multiplying (3.1) by w_{1} ∈ H^{1}(Ω_{1}) such that w_{1}(a) = 0 and integrating by parts using the first condition in (3.4) we obtain the identity:

[v_{1}, w_{1}]_{1} + α_{1}v_{1}(b) w_{1}(b) − β_{1}v_{2}(c) w_{1}(b) = λ (v_{1}, w_{1})_{L_{2} (Ω_{1} )} .

Analogously, multiplying (3.2) by w_{2} ∈ H^{1}(Ω_{2}) such that w_{2}(d) = 0 and integrating by parts we obtain:

[v_{2}, w_{2}]_{2} + α_{2}v_{2}(c) w_{2}(c) − β_{2}v_{1}(b) w_{2}(b) = λ (v_{2}, w_{2})_{L_{2} (Ω_{2} )} .

Now multiplying the first of these identities by β_{2}, the second by β_{1} and summing up we get (3.5). Conversely, consider the eigenpair [λ, v] of the variational spectral problem (3.5). Choosing successively test functions of the form w = (φ_{1}, 0) and w = (0, φ_{2}), φ_{i} ∈ C_{0}^{∞} (Ω_{i}), and using the standard rules of differentiation for Schwartz distributions [15, 22] we recover differential equations (3.1) and (3.2). To recover boundary conditions (3.4) we choose the test functions w = (w_{1}, 0) ∈ H^{1} and w = (0, w_{2}) ∈ H^{1}. □

We state the following important properties of the spaces H^{1} and L:

(i) H^{1} and L are Hilbert spaces, (ii) H^{1} is compactly embedded in L.

In the following lemma we deal with some properties of the bilinear form A(u, v).

Lemma 3.2. [11] Under condition (1.7), the bilinear form A, defined by (3.5), is symmetric and bounded on H^{1} × H^{1}. If conditions (1.8)–(1.9) also hold, the bilinear form A satisfies the Gårding inequality

A(v, v) ⩾ c_{1}∥v∥_{H^{1}}^{2} − c_{2}∥v∥_{L}^{2}, ∀ v ∈ H^{1}, c_{1}, c_{2} > 0.

Remark 3.1. If besides (1.7)–(1.9), conditions (1.10)–(1.11) are satisfied, the bilinear form A is coercive, i.e., there exists a constant c_{0} > 0 such that

A(v, v) ⩾ c_{0}∥v∥_{H^{1}}^{2} , ∀ v ∈ H^{1}.

Lemmas 2.1, 3.1, 3.2 and properties (i), (ii), allow us to recast problem (3.1)– (3.4) into the general theory of abstract eigenvalue problems for bilinear forms in Hilbert spaces, see e.g. [15, 18, 20]. This ensures the existence of exact eigenpairs as stated in the following theorem.

Theorem 3.1. Under conditions (1.7) and (1.8) problem (3.1)–(3.4) has a countable sequence of real eigenvalues −c_{2} < λ_{1} ⩽ λ_{2} ⩽ ⋯→ ∞. The corresponding eigenfunctions v^{k} ≡ (v_{1}^{k}, v_{2}^{k}), k = 1, 2 …, can be chosen to be orthonormal in L. They constitute a Hilbert basis for H^{1} as well as for L.

3.2. Existence and Uniqueness of Weak Solutions. Let us introduce the cylinders Q_{iT} = {(x, t) | x ∈ Ω_{i}, 0 < t < T }, i = 1, 2.

Theorem 3.2. Assume that f = (f_{1}, f_{2}) ∈ L_{2}(0, T; L), u_{0} = (u_{10}, u_{20}) ∈ L and

γ_{1}(t) = γ_{2}(t) = 0. Then problem (1.1)–(1.9) has a unique solution u = (u_{1}, u_{2}) ∈ H^{1,0} ≡ L_{2}(0, T; H^{1}) which satisfies the following weak formulation:

−β_{2} ∫_{Q_{1T}} u_{1} \frac{∂v_{1}}{∂t} dx dt − β_{1} ∫_{Q_{2T}} u_{2} \frac{∂v_{2}}{∂t} dx dt + ∫_{0}^{T} A(u(·, t), v(·, t)) dt

= β_{2} ∫_{Ω_{1}} u_{10}(x)v_{1}(x, 0) dx + β_{1} ∫_{Ω_{2}} u_{20}(x)v_{2}(x, 0) dx
+β_{2} ∫_{Q_{1T}} f_{1}(x, t)v_{1}dx dt + β_{1} ∫_{Q_{2T}} f_{2}(x, t)v_{2}dx dt ,

∀v = (v_{1}, v_{2}) ∈ H^{1,1} ≡ L_{2}(0, T; H^{1}) ∩ H^{1}(0, T; L), v_{i}(x, T) = 0 a.e. in Ω_{i}.

Proof. The existence proof, for example, can be accomplished by the Fourier method [15, 18, 20]. We first construct a family of approximate solutions, using the spectral problem (3.1)–(3.4). Since u_{i0} ∈ L_{2}(Ω_{i}), and f_{i}(x, t) ∈ L_{2}(Ω_{i}), i = 1, 2, for almost all t ∈ (0, T) we have

u_{i0}(x) = ∑_{k=1}^{∞} u_{i0}^{k}v_{i}^{k}(x), f_{i}(x, t) = ∑_{k=1}^{∞} f_{i}^{k}(t)v_{i}^{k}(x), i = 1, 2,

where u_{i0}^{k} = (u_{i0}, v_{i}^{k})_{L_{2} (Ω_{i} )} , f_{i}^{k}(t) = (f_{i}(·, t), v_{i}^{k})_{L_{2} (Ω_{i} )} and f_{i}^{k}(t) ∈ L_{2}(0, T). Let us consider for each k = 1, 2, … the functions

U_{i}^{k}(t) = u_{i0}^{k}e^{−λ_{k} t} + ∫_{0}^{t} f_{i}^{k}(τ)e^{λ_{k} (t−τ)} dτ, i = 1, 2,

which satisfy almost everywhere on (0, T) the equations

\frac{dU_{i}^{k}}{dt} + λ_{k}U^{k} = f_{i}^{k}(t), U_{i}^{k}(0) = u_{i0}^{k}, i = 1, 2.

The pair of sums S^{K} ≡ (S_{1}^{K} , S_{2}^{K}), S_{i}^{K} (x, t) = ∑ _{k=1}^{K} U_{i}^{k}(t)v_{i}^{k}(x), i = 1, 2, is a weak solution of the problem (1.1)–(1.9) with initial functions ∑ _{k=1}^{K} u_{i0}^{k}v_{i}^{k}(x) and right hand sides ∑ _{k=1}^{K} f_{i}^{k}v_{i}^{k}(x), i = 1, 2. Further, following the known techniques [15, 18, 20], we show that the series S^{K} is convergent in H^{1,0}.

Let u^{1}(x, t) ≡ (u_{1}^{1}(x, t), u_{2}^{1}(x, t)) and u^{2}(x, t) ≡ (u_{1}^{2}(x, t), u_{2}^{2}(x, t)) be two generalized solutions of problem (1.1)–(1.9). Then u = u^{1} − u^{2} is a generalized solution of the corresponding homogeneous problem. Let introduce the functions

v_{i}(x, t) = \{ \begin{matrix}∫_{t}^{l} u_{i}(x, τ) dτ, & x ∈ Ω_{i}, t ∈ (0, l), \\ 0, & x ∈ Ω_{i}, t ∈ (l, T), \end{matrix}

where i = 1, 2 and 0 < ε ⩽ l ⩽ T. It is easy to check that the functions v_{i} have generalized derivatives

\frac{∂v_{i}}{∂t} = \{ \begin{matrix}−u_{i}, & x ∈ Ω_{i}, t ∈ (0, l), \\ 0, & x ∈ Ω_{i}, t ∈ (l, T), \end{matrix}

\frac{∂v_{i}}{∂x} = \{ \begin{matrix}∫_{t}^{l} \frac{∂u_{i}}{∂x} (x, τ) dτ, & x ∈ Ω_{i}, t ∈ (0, l), \\ 0, & x ∈ Ω_{i}, t ∈ (l, T), \end{matrix}

v = (v_{1}, v_{2}) ∈ H^{1,1}, v_{i}(x, T) = 0, i = 1, 2, and v_{1}(a, t) = 0, v_{2}(d, t) = 0, t ∈ [0, T]. Plugging these formulas in the identity of Theorem 3.2, that defines the solution u, and assuming that u_{i}(x, t) = 0 for x ∈ Ω_{i}, t ∈ (0, l − ε)we get the equality

(3.6)  ∫_{l−ε}^{l} ∥u(·, t)∥_{L}^{2}dt + ∫_{l−ε}^{l} A(u(·, t), v(·, t)) dt = 0.

Following the technique from [18, p. 395], and using Lemma 3.2 we show that

(3.7) ∫_{l−ε}^{l} A(u(·, t), v(·, t)) dt = −\frac{1}{2} ∫_{l−ε}^{l} \frac{d}{dt}( A(v(·, t), v(·, t))) dt

= \frac{1}{2} A(v(·, l − ε), v(·, l − ε)) ⩾ \frac{c_{1}}{2} ∥v(·, l − ε)∥_{H^{1}}^{2} − \frac{c_{2}}{2} ∥v(·, l − ε)∥_{L}^{2}.

Further, we have

(3.8)  ∥v(·, l − ε)∥_{L}^{2} = ∑_{i=1}^{2} β_{3−i} ∫_{Ω_{i}} ( ∫_{l−ε}^{l} u_{i}(x, t) dt) ^{2}dx

⩽ ε ∑_{i=1}^{2} β_{3−i} ∫_{Ω_{i}} ∫_{l−ε}^{l} u_{i}^{2}(x, t) dt dx = ε ∫_{l−ε}^{l} ∥u(·, t)∥_{L}^{2} dt.

Choosing ε = T/N < 2/c_{2} from (3.6)–(3.8) follows ∫_{l−ε}^{l} ∥u(·, t)∥_{L}^{2} dt ⩽ 0, which imply that u_{i} = 0 a.e. in Ω_{i} × (l − ε, l), i = 1, 2.

Repeating this procedure for l = ε, 2ε, …, Nε = T we obtain that u_{i} = 0 a.e. in Q_{iT}, i = 1, 2, wherefrom follows u = 0 and u^{1} = u^{2}. □

3.3. A Priori Estimates and Energy Stability of the Solutions. Let H^{−1} = (H^{1})^{∗} be the dual space for H^{1}. The spaces H^{1}, L and H^{−1} form a Gelfand triple H^{1} ⊂ L ⊂ H^{−1} [22], with continuous and dense embeddings. We also introduce the space W = \{u | u ∈ L_{2}(0, T; H^{1}), \frac{∂u}{∂t} = ( \frac{∂u_{1}}{∂t} , \frac{∂u_{2}}{∂t} ) ∈ L_{2}(0, T; H^{−1} )\} with the inner product and associated norm

(u, v)_{W} = ∫_{0}^{T} [(u(·, t), v(·, t))_{H^{1}} + ( \frac{∂u}{∂t}(·, t), \frac{∂v}{∂t}(·, t))_{H^{−1}} ]dt, ∥u∥_{W} = (u, u)_{W}^{1/2} .

Let us consider the following weak form of (1.1)–(1.5):

(3.9)  ⟨ \frac{∂u}{∂t}(·, t), v⟩ + A(u(·, t), v) = ⟨f(·, t), v⟩, ∀v ∈ H^{1},

where ⟨·, ·⟩ denotes duality pairing in H^{−1} × H^{1}.

Problem (3.9) fit into the general theory of abstract parabolic problems [22]. Applying Theorem 26.1 from [22] to (3.9) we immediately obtain the following assertion.

Theorem 3.3. Let the assumptions (1.7)–(1.9) hold and suppose that u_{0} = (u_{10}, u_{20}) ∈ L, f = (f_{1}, f_{2}) ∈ L_{2}(0, T; H^{−1} ), γ_{1}(t) = γ_{2}(t) = 0. Then the problem (1.1)–(1.6) has a unique weak solution u ∈ W, and Hadamard’s a priori estimate holds:

∥u∥_{W}^{2} ⩽ C(T)( ∥u_{0}∥_{L}^{2} + ∫_{0}^{T} ∥f(·, t)∥_{H^{−1}}^{2} dt) , C(T) = Ce^{2c_{2} T} .

Theorem 3.4. Let the assumptions (1.7)–(1.11) hold and suppose that u_{0} = (u_{10}, u_{20}) ∈ L, f = (f_{1}, f_{2}) ∈ L_{2}(0, T; L), γ_{1}(t) = γ_{2}(t) = 0. Then the solution of problem (1.1)–(1.6) satisfies the a priori estimate:

∥u(·, t)∥_{L}^{2} ⩽ e^{−2δt} ( ∥u_{0}∥_{L}^{2} + C ∫_{0}^{t} e^{2δτ} ∥f(·, τ)∥_{L}^{2} dτ) , δ > 0.

Proof. Setting in (3.9) v = u and using Theorem 3.1, and the Cauchy– Schwarz inequality with ε > 0 one obtains

\frac{1}{2} \frac{∂}{∂t}( ∥u(·, t)∥_{L}^{2}) + λ_{1} ∥u(·, t)∥_{L}^{2} ⩽ ε ∥u(·, t)∥_{L}^{2} + \frac{1}{4ε} ∥f(·, t)∥_{L}^{2},

where λ_{1} > 0 is the minimal eigenvalue of the problem (3.1)–(3.4). For ε < λ_{1}, result with δ = λ_{1} − ε and C = 1/(2ε) follows by integration. □

Remark 3.2. Functional ∥u(·, t)∥_{L}^{2} express the kinetic energy of the system. Therefore, results of such type are usually referred to as energy stability (cf. [9]).

Remark 3.3. The minimal eigenvalue of problem (3.1)–(3.4) may be positive even if (1.10)–(1.11) is not satisfied. Using Lemma 3.2 and the Poincaré inequality one obtains

A(v, v) ⩾ c_{3} ∥v∥_{L}^{2}, c_{3} = \frac{2c_{1}}{max\{(b − a)^{2}, (d − c)^{2}\}} + c_{1} − c_{2}.

In such a way, instead of (1.10)–(1.11) we can require that c_{3} > 0.

4. Nonnegativity Preservation

Considering the heat conduction problem, the temperature of a body can not be negative if the temperature is nonnegative in the initial state and on the boundary of the body. This property is called nonnegativity preservation. In this section we analyze the nonnegativity preservation for the problem (1.1)–(1.11).

For a function v_{i} ∈ H^{1}(Ω_{i}), i = 1, 2, we denote its positive and negative parts by v_{i}^{+} and v_{i}^{−} . That is v_{i} = v_{i}^{+} + v_{i}^{−} , where v_{i}^{+} = max{v_{i}, 0} ⩾ 0 and v_{i}^{−} = min{v_{i}, 0} ⩽ 0. By [6, Lemma 7.6, p. 150], we have that

\frac{dv_{i}^{+}}{dx} = \{ \begin{matrix}\frac{dv_{i}}{dx} , & if v ⩾ 0, \\ 0, & if v ⩽ 0, \end{matrix} \frac{dv_{i}^{−}}{dx} = \{ \begin{matrix}\frac{dv_{i}}{dx} , & if v ⩽ 0, \\ 0, & if v ⩾ 0. \end{matrix}

It follows that

v_{i}^{+} v_{i}^{−} = \frac{dv_{i}^{+}}{dx} \frac{dv_{i}^{−}}{dx} = v_{i}^{−} \frac{dv_{i}^{+}}{dx} + v_{i}^{+} \frac{dv_{i}^{−}}{dx} = 0 a.e. in Ω_{i}, i = 1, 2.

For some T > 0 let us consider the following inequalities

\frac{∂u_{i}}{∂t} + L_{i}u_{i} ⩾ 0 in Q_{iT} , i = 1, 2, (4.1)

l_{1}(u_{1}, u_{2}) ⩾ 0, l_{2}(u_{1}, u_{2}) ⩾ 0, 0 < t < T. (4.2)

Let V (see [16, p. 6]) be the Banach space consisting of all functions in H^{1,0} having the finite norm

∥u∥_{V} = max_{i}( sup_{0⩽t⩽T} ∥u_{i}(·, t)∥_{L_{2} (Ω_{i} )} + ∥ \frac{∂u_{i}}{∂x} ∥_{L_{2} (Q_{iT} )} ) .

A function u = (u_{1}, u_{2}) ∈ V is said to satisfy weakly (4.1) and (4.2) if for any v = (v_{1}, v_{2}) ∈ H^{1,1}, v_{i} ⩾ 0, i = 1, 2:

(4.3) \begin{aligned}β_{2} ∫_{a}^{b}u_{1}(x, t)v_{1}(x, t) dx + β_{1} ∫_{c}^{d} & u_{2}(x, t)v_{2}(x, t) dx \\ −β_{2} ∫_{a}^{b}u_{1}(x, 0)v_{1}(x, 0) dx − β_{1} ∫_{c}^{d} & u_{2}(x, 0)v_{2}(x, 0) dx \\ −β_{2} ∫_{0}^{t}∫_{a}^{b}u_{1}(x, τ)\frac{∂v_{1}}{∂t} (x, τ) dx dτ & − β_{1} ∫_{0}^{t}∫_{c}^{d}u_{2}(x, τ)\frac{∂v_{2}}{∂t} (x, τ) dx dτ \\ & + ∫_{0}^{t} A(u(·, τ), v(·, τ)) dτ ⩾ 0 \end{aligned}

for almost every t ∈ (0, T). It is easy to see that if u_{i} ∈ C^{2,1}(\overline{Q}_{iT}), i = 1, 2, satisfy (4.1), (4.2) pointwise then (4.3) holds. Conversely, if u ∈ V satisfy (4.3) and u_{i} are sufficiently smooth, then they also satisfy (4.1), (4.2) pointwise. We shall now prove the following theorem.

Theorem 4.1. Let u = (u_{1}, u_{2}) ∈ V satisfy (4.1) and (4.2) weakly, and let (1.7)–(1.11) hold. If u_{i}(x, 0) ⩾ 0, then u_{i}(x, t) ⩾ 0 a.e. in Q_{iT}, i = 1, 2.

Proof. Using the Steklov average and passing to the limit (see [6, Ch. 3]) we can formally take v = −u^{−} ⩾ 0 in (4.3) to obtain

(4.4) \begin{aligned}\frac{β_{2}}{2} ∫_{a}^{b}(u_{1}^{−} (x, t))^{2}dx & + \frac{β_{1}}{2} ∫_{c}^{d}(u_{2}^{−} (x, t))^{2}dx − \frac{β_{2}}{2} ∫_{a}^{b}(u_{1}^{−} (x, 0))^{2}dx \\ −\frac{β_{1}}{2} ∫_{c}^{d}(u_{2}^{−} (x, 0))^{2}dx & + β_{2} ∫_{0}^{t}∫_{a}^{b}( p_{1}(x)\frac{∂u_{1}}{∂x} \frac{∂u_{1}^{−}}{∂x} + q_{1}(x)u_{1}u_{1}^{−} ) dx dt \\ & + β_{1} ∫_{0}^{t}∫_{c}^{d} ( p_{2}(x)\frac{∂u_{2}}{∂x} \frac{∂u_{2}^{−}}{∂x} + q_{2}(x)u_{2}u_{2}^{−} ) dx dt \\ ⩽ ∫_{0}^{t} \{ − β_{2} & α_{1}u_{1}(b, τ)u_{1}^{−} (b, τ) − β_{1}α_{2}u_{2}(c, τ)u_{2}^{−} (c, τ) \\ & + β_{1}β_{2}[u_{1}(b, τ)u_{2}^{−} (c, τ) + u_{2}(c, τ)u_{1}^{−} (b, τ)]\} dτ \end{aligned}

Further

\frac{∂u_{i}}{∂x} \frac{∂u_{i}^{−}}{∂x} = ( \frac{∂u_{i}^{−}}{∂x} ) ^{2} , (u_{i}u_{i}^{−} ) = (u_{i}^{−} )^{2},

\begin{aligned}u_{1}(b, t)u_{2}^{−} (c, t) + u_{2}(c, t)u_{1}^{−} (b, t) & = 2u_{1}^{−} (b, t)u_{2}^{−} (c, t) + u_{1}^{+} (b, t)u_{2}^{−} (c, t) \\ & + u_{1}^{−} (b, t)u_{2}^{+} (c, t) ⩽ 2u_{1}^{−} (b, t)u_{2}^{−} (c, t). \end{aligned}

From here and (4.4), since by our assumption u_{i}^{−} (x, 0) = 0, i = 1, 2, we obtain:

\frac{β_{2}}{2} ∫_{a}^{b}(u_{1}^{−} (x, t))^{2}dx + β_{2} ∫_{0}^{t}∫_{a}^{b}( p_{1}(x)( \frac{∂u_{1}^{−}}{∂x} )^{2} + q_{1}(x)(u_{1}^{−} )^{2}) dx dτ

+\frac{β_{1}}{2} ∫_{c}^{d}(u_{2}^{−} (x, t))^{2}dx + β_{1} ∫_{0}^{t}∫_{c}^{d} ( p_{2}(x)( \frac{∂u_{2}^{−}}{∂x} )^{2} + q_{2}(x)(u_{2}^{−} )^{2}) dx dτ

⩽ − ∫_{0}^{t} [β_{2}α_{1}(u_{1}^{−} (b, τ))^{2} + β_{1}α_{2}(u_{2}^{−} (c, τ))^{2} − 2β_{1}β_{2}u_{1}^{−} (b, τ)u_{2}^{−} (c, τ)]d τ ⩽ 0.

For the last inequality we used (1.9) and (1.11). From here it follows u_{i}^{−} (x, t) = 0. We then conclude that u_{i}(x, t) = u_{i}^{+} (x, t) ⩾ 0, a.e. in Q_{iT}, i = 1, 2. □

5. Concluding Remarks

We have proposed a simple parabolic problem on disjoint intervals to model layers with material properties which significantly differ from those of the surrounding medium. We assumed that the differential equation that describe the process in the layer is monotone, i.e., it satisfies the maximum principle which implies the positivity conditions (1.9). Under some additional natural assumptions on the input data we proved well-posedness of the IBVP that describes the process outside the layer as well as nonnegativity and energy stability of the solutions of the problem. The results obtained in this paper should be a good base for analysis of numerical approximations of the problems. The preservation of characteristic properties of different phenomena is a very important requirement in the construction of reliable numerical methods [5]. The nonnegativity preservation of finite element solutions to 1D elliptic problem on disjoint domains is discussed in [13]. A further development of the present modelling of layer problems should include cases when the differential operator corresponding to the process in the layer is nonmonotonous and conditions (1.9) fail. The process in the intermediate layer is considered to be stationary (Section 2) and as a result of this the derived nonlocal boundary conditions are of the Robin–Dirichlet type [7, 8]. But when the process in the intermediate layer is nonstationary, the arising boundary conditions will be nonlocal in time and it will be interesting to study this new problem for well-posedness and qualitative behavior of the solutions. Nonlinear 1D elliptic problems on disjoint domains are studied in [21]. Using the exact solutions of heat mass transfer equations [19] the results from the present paper could be extended to nonlinear parabolic problems.

Acknowledgement

The research of the first author was supported by Ministry of Education and Science of Republic of Serbia under project 174015. The research of the second author was supported by Bulgarian Fund of Science under Projects ID-09-0186 and Bg-Sk-203.

References

[1] L. Caffarelli, A monotonicity formula for heat functions in disjoint domains, in: B. V. Problems for PDEs and Appls., RMA Res. Notes Appl. Math. 29, 53-60, Masson, Paris, 1993.

[2] F. Calabro, P. Zunino, Analysis of parabolic problems on partitioned domains with nonlinear conditions at the interface. Application to mass transfer trough semi-permeable membranes, Math. Models and Methods in Appl. Sci. 164 (2006), 479–501.

[3] H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, Oxford Univ. Press, 2000.

[4] A. K. Datta, Biological and Bioenvironmental Heat and Mass Transfer, Marcel Dekker, 2002.

[5] I. Farago, R. Horvath, On the nonnegativity conservation of finite element solutions of parabolic problems, CAKUTO Int. Ser. Math. Sci. Appl. 15 (2001), 76–84.

[6] D. Gilbarg, N. Truddinger, Elliptic Partial Differential Equations of Second Order, 3rd edition, Springer, 2001.

[7] D. Givoli, Exact representation on artificial interfaces and applications in mechanics, Appl. Mech. Rev. 52 (1999), 333–349.

[8] D. Givoli, Finite element modeling of thin layers, CMES 5, No 6 (2004), 497–514.

[9] B. S. Jovanovic, L. G. Vulkov, Energy stability for a class of two-dimensional interface linear parabolic problems, J. Math. Anal. Appl. 311 (2005), 120–138.

[10] B. S. Jovanović, L. G. Vulkov, Finite difference approximation of strong solutions of a parabolic interface problem on disconnected domains, Publ. Inst. Math., Nouv. Sér. 84(98), (2008), 37–48.

[11] B. S. Jovanović, L. G. Vulkov, Numerical solution of a parabolic transmission problem, IMA J. Numer. Anal. 31 (2011), 233–253.

[12] H. Kalis, I. Kangro, Increasing of accuracy for engineering calculation of heat transfer problems in two layer media, Mathematical Modelling and Analysis, 10:2 (2005), 173–190.

[13] M. Koleva, Finite element solutions of 1D boundary value linear and nonlinear problems with nonlocal jump conditions; in: M. Todorov (Ed.), Amer. Inst. of Phys. Conf. Proceed. Series 964 (2007), 163–170.

[14] V. I. Korzyuk, S. V. Lemeshevsky, P. P. Matus, V. N. Shalima, Conjugation problem about jointly separate flow of viscoelastic and viscous fluids in the plane duct, Mathematical Modelling and Analysis, 4:1 (1999), 114–123.

[15] J. L. Lions, E. Magenes, Non Homogeneous Boundary Value Problems and Applications, Springer-Verlag, Berlin and New York, 1972.

[16] O. A. Ladyzenskaya, V. A. Solonnikov, N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type, in: Amer. Math. Soc. Transl. Monographs, Vol. 23, 1968.

[17] A. V. Lykov, Heat-Mass Transfer, Energiya, Moscow, 1978 (in Russian).

[18] V. P. Mikhailov, Partial Differential Equations, Nauka, Moscow, 1980 (in Russian).

[19] A. D. Polyanin, A. I. Zhurov, Exact solutions of heat and mass transfer equations, Matematica Contemporanea 19 (2000), 105–127.

[20] M. Renardy, R. C. Rogers, An Introduction to Partial Differential Equations, Springer-Verlag, Berlin, 1993.

[21] L. G. Vulkov, Well posedness and a monotone iterative method for a nonlinear interface problem on disjoint intervals, in: M. Todorov (Ed.), Amer. Inst. of Phys. Conf. Proceed. Series 964 (2007), 188–195.

[22] J. Wloka, Partial Differential Equations, Cambridge Univ. Press, 1987.

University of Belgrade (Received 07 11 2011) Faculty of Mathematics (Revised 21 02 2012) 11000 Belgrade, Serbia

    bosko@matf.bg.ac.rs

University of Rousse

Department of Mathematics

7017 Rousse, Bulgaria

    lvalkov@uni-ruse.bg