Electronic Journal of Qualitative Theory of Differential Equations Proc. 8th Coll. QTDE, 2008, No. 8 1-15; http://www.math.u-szeged.hu/ejqtde/

Qualitative Properties of Monotone Linear Parabolic Operators^{∗}^{†}

István Faragó^{‡}  Róbert Horváth^{§}

Abstract

When we construct mathematical or numerical models in order to solve a real-life problem, it is important to preserve the characteristic properties of the original process. The models have to possess the equivalents of these properties. Parabolic partial equations generally serve as the mathematical models of heat conduction or diffusion processes, where the most important properties are the monotonicity, the nonnegativity preservation and the maximum principles. In this paper, the validity of the equivalents of these qualitative properties are investigated for the second order linear partial differential operator. Conditions are given that guarantee the qualitative properties. On some examples we investigate these conditions.

Keywords: qualitative properties, monotone operators, maximum principles, parabolic problems. Mathematical Subject Classification: 35B50, 35K10.

∗This paper is in final form and no version of it is submitted for publication elsewhere.

†The authors were supported by the National Scientific Research Found (OTKA)

K67819, K61800, K49819.

‡Eötvös Loránd University, Hungary, Department of Applied Analysis and Computational Mathematics, H-1117 Budapest, Pázmány P. sétány 1/C., E-mail:

faragois@cs.elte.hu.

§University of West-Hungary, Hungary, Institute of Mathematics and Statistics, H-9400 Sopron, Erzsébet u. 9., E-mail: rhorvath@ktk.nyme.hu.

1 Introduction, motivation

The classical theory of partial differential equations investigates the question of existence, uniqueness and the methods that produce the solutions of the equations. Qualitative investigations came into being from the mid-fifties. Researchers assumed that the solution of the problem is at hand and tried to answer the questions: What kind of properties does the solution have? What kind of class of functions does the solution belong to? The most representative result in this field is the well-known maximum principle. A comprehensive survey of the qualitative properties of the second order linear partial differential equations can be found e.g. in [2, 7].

Real-life phenomena possess a number of characteristic properties. For instance, let us consider the heat conduction of a physical body. When we increase the strength of the heat sources inside the body, the temperature on the boundary and the temperature in the initial state, then the temperature must not decrease inside the body. This property is called monotonicity. Maximum principles result in lower and upper bounds for the distribution of temperature in the body. One of the simplest form of them states that, if there are no heat sources and sinks present inside the body, then the maximum temperature appears also on the boundary of the body or in the initial state. The above example shows that when we construct a mathematical model of a phenomenon, it is important to investigate whether the mathematical model possesses the same properties as the original process. In this paper we investigate the validity of the monotonicity property and one of the maximum principles for the second order linear parabolic partial differential operator, and reveal the connections of the properties. The results of the qualitative theory of differential equations, albeit they have the importance on their own, help us to show that the qualitative properties of a mathematical model correspond to the qualitative properties of the modelled phenomenon.

The qualitative adequateness can be investigated also for numerical models, but this topic is beyond the scope of this paper. For more details consult e.g. [1, 4, 5].

The paper is organized as follows. In Section 2, we start with the general investigation of the qualitative properties of linear operators and we reveal their relations. The results are applied to second order parabolic differential operators in Section 3. Conditions are given that guarantee the qualitative properties. On some examples we investigate these conditions.

2 Qualitative Properties of Linear Operators

Let S, S_{∂} be two arbitrary mutually disjoint sets in IR^{d} (1 ≤ d ∈ IN) and let T be an arbitrary positive number. Let X and Y denote given ordered vector spaces of bounded real valued functions, or some subspaces of them, defined on (S ∪ S_{∂}) × [0, T] and S × (0, T), respectively. We introduce the notations

K_{τ} = S × (0, τ),  K̄_{τ} = (S ∪ S_{∂}) × [0, τ],
Kτ̄ = S × (0, τ], G_{τ} = (S_{∂} × [0, τ]) ∪ (S × {0})

for any arbitrary positive number τ.

Let L : X → Y be a linear operator. Next we define some important qualitative properties of L.

Definition 2.1. The operator L is said to be monotone if for all t^{⋆} ∈ (0, T) and v_{1}, v_{2} ∈ X such that v_{1}|_{G_{t^{⋆}}} ≥ v_{2}|_{G_{t^{⋆}}} and (Lv_{1})|_{Kt̄^{⋆}} ≥ (Lv_{2})|_{Kt̄^{⋆}}, the relation v_{1}|_{Kt̄^{⋆}} ≥ v_{2}|_{Kt̄^{⋆}} holds.

Notice that the monotonicity property gives possibility to compare two elements on the entire Kt̄^{⋆}, knowing their image relation on this set and their relation on G_{t^{⋆}}. This phenomenon is generally called comparison principle. From the practical point of view, the comparison principle is used for comparison of an unknown function with some other known function. On the base of the comparison principle, we can formulate some further qualitative properties.

Definition 2.2. The operator L is said to be nonnegativity preserving if the relations v|_{G_{t^{⋆}}} ≥ 0 and (Lv)|_{Kt̄^{⋆}} ≥ 0 imply that the relation v|_{Kt̄^{⋆}} ≥ 0 also holds.

Clearly, the nonnegativity preservation property means the comparison of the function v_{1} = v with the function v_{2} = 0. Moreover, we have

Corollary 2.3 For linear operators L : X → Y, the monotonicity and the nonnegativity preserving properties are equivalent.

When an unknown function v can be compared with a function that is given by the values (Lv)|_{Kt̄^{⋆}} and v|_{G_{t^{⋆}}}, then we say that a maximum principle is defined. In the following we define four types of maximum principles, which differ in the definition of the comparison functions. More precisely, these comparison functions are constructed via sup_{G_{t^{⋆}}} v and sup_{Kt̄^{⋆}} Lv.

Definition 2.4. We say that the operator L satisfies the weak maximum principle if for any function v ∈ X and t^{⋆} ∈ (0, T) the inequality

sup_{K̄_{t^{⋆}}} v ≤ max{0, sup_{G_{t^{⋆}}} v} + t^{⋆} · max{0, sup_{Kt̄^{⋆}} Lv}  (1)

is satisfied.

Definition 2.5. We say that the operator L satisfies the strong maximum principle if for any function v ∈ X and t^{⋆} ∈ (0, T) the inequality

sup_{K̄_{t^{⋆}}} v ≤ sup_{G_{t^{⋆}}} v + t^{⋆} · max{0, sup_{Kt̄^{⋆}} Lv}  (2)

is satisfied.

When the sign of Lv is known, then the comparison function is constructed only via sup_{G_{t^{⋆}}} v. Such type of maximum principles are called boundary maximum principles (c.f. [11]).

Definition 2.6. We say that the operator L satisfies the weak boundary maximum principle when for any function v ∈ X and t^{⋆} ∈ (0, T) such that Lv|_{Kt̄^{⋆}} ≤ 0 the inequality

sup_{K̄_{t^{⋆}}} v ≤ max{0, sup_{G_{t^{⋆}}} v}  (3)

holds.

Definition 2.7. We say that the operator L satisfies the strong boundary maximum principle when for any function v ∈ X and t^{⋆} ∈ (0, T) such that Lv|_{Kt̄^{⋆}} ≤ 0 the equality

sup_{K̄_{t^{⋆}}} v = sup_{G_{t^{⋆}}} v  (4)

holds.

Remark 2.8 Because v ∈ X implies −v ∈ X and the supremum of a real valued function v is minus one times the infimum of −v, the maximum principles can be written in an equivalent forms with minimums and infimums.

Remark 2.9 Assume that v has maximum (minimum) on K̄_{t^{⋆}}. Then, in case of validity of any boundary maximum principle, we can give the location of the maximum (minimum): it is taken on G_{t^{⋆}}.

As a first step, we investigate the relations of the above properties.

Theorem 2.10 For a linear operator L : X → Y, the following implications are valid

i) the strong maximum principle implies the strong boundary maximum principle,

ii) the strong boundary maximum principle implies the weak boundary maximum principle,

iii) the weak maximum principle implies the weak boundary maximum principle,

iv) the weak boundary maximum principle implies the monotonicity of the operator.

Proof. In Implication i), the fact that the right-hand side of (4) is not greater than the left-hand side is obvious. The reverse relation follows from (2) and the non-positivity of Lv on Kt̄^{⋆}.

Implication ii) follows from the trivial relation max{0, sup_{G_{t^{⋆}}} v} ≥ sup_{G_{t^{⋆}}} v. Implication iii) is trivial due to the non-positivity of Lv on Kt̄^{⋆}. In order to prove Implication iv), let v_{1}, v_{2} ∈ X be two arbitrary functions

and t^{⋆} ∈ (0, T) an arbitrary real number with the properties v_{1}|_{G_{t^{⋆}}} ≥ v_{2}|_{G_{t^{⋆}}} and (Lv_{1})|_{Kt̄^{⋆}} ≥ (Lv_{2})|_{Kt̄^{⋆}}. Then, using the linearity of the operator L, we obtain that (L(v_{2}−v_{1}))|_{Kt̄^{⋆}} ≤ 0. It follows from the weak boundary maximum principle that

sup_{K̄_{t^{⋆}}}(v_{2} − v_{1}) ≤ max{0, sup_{G_{t^{⋆}}}(v_{2} − v_{1})} = 0,

which results in the relation v_{1}|_{Kt̄^{⋆}} ≥ v_{2}|_{Kt̄^{⋆}}. This completes the proof.

Corollary 2.11 If a linear operator L : X → Y satisfies any of the maximum principles, then it is monotone.

Now we analyze the reverse implication: Under which conditions will the monotonicity imply the maximum principles? Because it causes no confusion, we denote the constant one function simply by 1 and the function (x, t) ↦ t by t. These functions are supposed to be defined on K_{T}.

Theorem 2.12 Let us suppose that the space X contains the functions 1 and t. Then, if a monotone operator L has the properties L1 ≥ 0 and Lt ≥ 1, then L possesses both the weak maximum principle and the weak boundary maximum principle.

Proof. Because of Implication iii) in Theorem 2.10, it is enough to prove the validity of the weak maximum principle. Let t^{⋆} be a fixed number from the interval (0, T). We choose an arbitrary function v_{2} ∈ X and we set v_{1} = max{0, sup_{G_{t^{⋆}}} v_{2}} + t · max{0, sup_{Kt̄^{⋆}} Lv_{2}}, which function trivially belongs to X, because 1, t ∈ X. Clearly, v_{1}|_{G_{t^{⋆}}} ≥ v_{2}|_{G_{t^{⋆}}}, moreover

Lv_{1} = max{0, sup_{G_{t^{⋆}}} v_{2}}(L1) + max{0, sup_{Kt̄^{⋆}} Lv_{2}}(Lt) ≥ sup_{Kt̄^{⋆}} Lv_{2} ≥ Lv_{2}

on Kt̄^{⋆}. Hence, based on the monotonicity of the operator, we obtain that

v_{1} = max{0, sup_{G_{t^{⋆}}} v_{2}} + t · max{0, sup_{Kt̄^{⋆}} Lv_{2}} ≥ v_{2}

on Kt̄^{⋆}. This completes the proof.

Theorem 2.13 Let us suppose that the space X contains the functions 1 and t. Then, if a monotone operator L has the properties L1 = 0 and Lt ≥ 1, then L possesses all the investigated maximum principles.

Proof. Because of Theorem 2.10 and Theorem 2.12, it is enough to prove the validity of the strong maximum principle. Let t^{⋆} be any fixed number from the interval (0, T). We choose an arbitrary function v_{2} ∈ X and we set v_{1} = sup_{G_{t^{⋆}}} v_{2} + t · max{0, sup_{Kt̄^{⋆}} Lv_{2}}. Clearly, v_{1}|_{G_{t^{⋆}}} ≥ v_{2}|_{G_{t^{⋆}}}, moreover

Lv_{1} = (sup_{G_{t^{⋆}}} v_{2})(L1) + max{0, sup_{Kt̄^{⋆}} Lv_{2}}(Lt) ≥ sup_{Kt̄^{⋆}} Lv_{2} ≥ Lv_{2}

on Kt̄^{⋆}. Based on the monotonicity of the operator we obtain that

v_{1} = sup_{G_{t^{⋆}}} v_{2} + t · max{0, sup_{Kt̄^{⋆}} Lv_{2}} ≥ v_{2}

on Kt̄^{⋆}. This completes the proof.

3 Qualitative Properties of the Second Order Parabolic Operator

In this section, the results obtained for the general linear operators are applied to the second order linear partial differential operator.

Let Ω and ∂Ω denote, respectively, a bounded domain in IR^{d} (1 ≤ d ∈ IN) and its boundary, and we introduce the sets

Q_{τ} = Ω × (0, τ),  Q̄_{τ} = Ω̄ × [0, τ],
Qτ̄ = Ω × (0, τ], Γ_{τ} = (∂Ω × [0, τ]) ∪ (Ω × {0})

for any arbitrary positive number τ. In the sequel, Γ_{τ} is called parabolic boundary. For some fixed number T > 0, we consider the second order linear partial differential operator

L ≡ \frac{∂}{∂t} − ∑_{m,k=1}^{d} a_{m,k} \frac{∂^{2}}{∂x_{m}∂x_{k}} − ∑_{m=1}^{d} a_{m} \frac{∂}{∂x_{m}} − a_{0},  (5)

where the coefficient functions are defined and bounded on Q_{T}. We assume that the operator is parabolic, that is the matrix

S(x, t) := [a_{m,k}(x, t)]_{m,k=1}^{d}  (6)

is positive definite at all points of Q_{T}. We define the domain of the operator L, denoted by dom L, as the space of functions v ∈ C(Q̄_{T}), for which the derivatives ∂v/∂x_{m}, ∂^{2}v/∂x_{m}∂x_{k} and ∂v/∂t exist in Q_{T} and they are bounded. It can be seen easily that Lv is bounded on Qt̄^{⋆} for each v ∈ dom L and t^{⋆} ∈ (0, T), which means that inf_{Qt̄^{⋆}} Lv and sup_{Qt̄^{⋆}} Lv are finite values.

The monotonicity and the maximum principles for operator (5) can be defined using the definitions of the previous section with the setting S = Ω, S_{∂} = ∂Ω, X = dom L and

Y = {w ∈ B(Q_{T}) | there exists v ∈ X such that Lv = w in Q_{T} },

where B(Q_{T}) denotes the space of bounded functions on Q_{T}. The fact G_{t^{⋆}} = Γ_{t^{⋆}} motivates the earlier phrase ”boundary” in the definitions of the previous section.

Remark 3.1 For the sake of simplicity, we used the terminology strong maximum principle, but distinction should be made between this property and the following one (called also strong maximum principle in the literature): if Lv ≤ 0 in Q_{T} and v assumes its positive maximum at an interior point (x^{0}, t^{0}), then v ≡ const. in the set of all points (x, t) ∈ Q_{T} which can be connected to (x^{0}, t^{0}) by a simple continuous curve in Q_{T} along which the coordinate t is non-decreasing from (x, t) to (x^{0}, t^{0}) ([10]). This property was extended also for boundary points in [6].

Remark 3.2 When operator (5) appears in a mathematical model of a physical phenomenon, then the physical units of the quantities must agree in the maximum principles. Now we check this agreement for the one-dimensional heat conduction operator. We use the units of SI, that is K=Kelvin, kg=kilogram, s=second, m=meter and J=Joule. Thus, let us consider the operator

L = \frac{∂}{∂t} − \frac{κ}{cϱ} \frac{∂^{2}}{∂x^{2}} ,

where κ is the heat conduction coefficient (measured in Jm/(Ks), ϱ is the density (measured in kg/m), c is the specific heat (measured in J/(kgK)) and the function v, which the operator is applied to, is the temperature (measured in K). Because the temperature is estimated in the maximum principles from above, to the agreement of the physical units we need to check that the quantity t sup_{Kt̄^{⋆}} Lv can be measured in Kelvin. Indeed, the unit of this quantity results in

s · (\frac{K}{s} − \frac{\frac{J}{Ks}}{\frac{Jm}{kgK} \frac{kg}{m}} \frac{K}{m^{2}} ) = K.

We turn to the investigation of the qualitative properties of operator (5). First we prove a basic property of operator L.

Theorem 3.3 Operator (5) is monotone.

Proof. We prove the statement in view of Corollary 2.3.

Let v ∈ dom L an arbitrary fixed function. Then the function

v̂(x, t) ≡ v(x, t)e^{−λt}  (7)

also belongs to dom L for any real parameter λ. Expressing v from (7) and applying operator (5) to it, we get

Lv = L(e^{λt}v̂) = e^{λt} [\frac{∂v̂}{∂t} − ∑_{m,k=1}^{d} a_{m,k} \frac{∂^{2}v̂}{∂x_{m}∂x_{k}} − ∑_{m=1}^{d} a_{m} \frac{∂v̂}{∂x_{m}} + (λ − a_{0})v̂] .

(8)

Let us fix the parameter t^{⋆} ∈ (0, T). Since v̂ is a continuous function on Q̄_{t^{⋆}}, its minimum exists and it is taken at some point (x^{0}, t^{0}) ∈ Q̄_{t^{⋆}}.

• First we assume that this point belongs to the parabolic boundary, i.e., (x^{0}, t^{0}) ∈ Γ_{t^{⋆}}. Then, due to the obvious relation

v̂(x, t) ≥ v̂(x^{0}, t^{0}) = min_{Γ_{t^{⋆}}} v̂

for all (x, t) ∈ Q̄_{t^{⋆}}, we get the estimation

inf_{Qt̄^{⋆}} v̂ ≥ min_{Γ_{t^{⋆}}} v̂.  (9)

• Assume now that (x^{0}, t^{0}) ∈ Qt̄^{⋆}. Then we get the relations

\frac{∂v̂}{∂t}(x^{0}, t^{0}) ≤ 0, \frac{∂v̂}{∂x_{m}} (x^{0}, t^{0}) = 0,  (10)

and, because (x^{0}, t^{0}) is a minimum point, the second derivative matrix

V̂(x^{0}, t^{0}) := [\frac{∂^{2}v̂}{∂x_{m}∂x_{k}} (x^{0}, t^{0})]_{m,k=1}^{d}

is positive semi-definite.

Let us denote the Hadamard product

(S(x^{0}, t^{0}) ∘ V̂(x^{0}, t^{0}))_{m,k} = a_{m,k}(x^{0}, t^{0}) · \frac{∂^{2}v̂}{∂x_{m}∂x_{k}} (x^{0}, t^{0})  (11)

(m, k = 1, …, d) by S(x^{0}, t^{0}) ∘ V̂(x^{0}, t^{0}) ∈ IR^{d×d}. Due to the assumptions, both the matrices S(x^{0}, t^{0}) and V̂(x^{0}, t^{0}) are positive semidefinite, hence, according to the Schur theorem (e.g. Theorem 7.5.3 in [8]), the matrix S(x^{0}, t^{0}) ∘ V̂(x^{0}, t^{0}) is also positive semi-definite.

We investigate (8) in the rearranged form

e^{−λt}Lv + ∑_{m=1}^{d} a_{m} \frac{∂v̂}{∂x_{m}} − (λ − a_{0})v̂ = \frac{∂v̂}{∂t} − ∑_{m,k=1}^{d} a_{m,k} \frac{∂^{2}v̂}{∂x_{m}∂x_{k}} . (12)

Introducing the notation e = [1, 1, …, 1]^{⊤} ∈ IR^{d}, the relation

∑_{m,k=1}^{d} a_{m,k}(x^{0}, t^{0})\frac{∂^{2}v̂}{∂x_{m}∂x_{k}} (x^{0}, t^{0}) = ((S(x^{0}, t^{0}) ∘ V̂(x^{0}, t^{0}))e, e) ≥ 0.

(13)

is valid. On the base of (10) and (13), the right-hand side of (12) is nonpositive at the point (x^{0}, t^{0}). Hence, the inequality

e^{−λt^{0}}(Lv)(x^{0}, t^{0}) − (λ − a_{0}(x^{0}, t^{0}))v̂(x^{0}, t^{0}) ≤ 0  (14)

holds. Let us introduce the notations a_{inf} := inf_{Q_{T}} a_{0} and a_{sup} := sup_{Q_{T}} a_{0}, which are well-defined because of the boundedness of the coefficient function a_{0}. For any λ > a_{sup}, we have

v̂(x^{0}, t^{0}) ≥ \frac{e^{−λt^{0}}(Lv)(x^{0}, t^{0})}{λ − a_{0}(x^{0}, t^{0})} ≥ \frac{e^{−λt^{0}}(Lv)(x^{0}, t^{0})}{λ − a_{inf}} ≥

≥ \frac{1}{λ − a_{inf}} inf_{Qt̄^{⋆}}(e^{−λt}(Lv)(x, t)).

(15)

Since the function v̂ takes its minimum at the point (x^{0}, t^{0}), therefore the estimation (15) shows the validity of the inequality

inf_{Qt̄^{⋆}} v̂ ≥ \frac{1}{λ − a_{inf}} inf_{Qt̄^{⋆}}(e^{−λt}(Lv)(x, t)).  (16)

Clearly, the estimates of the two different cases, namely (9) and (16) together, imply that

inf_{Q̄_{t^{⋆}}} v̂ ≥ min\{inf_{Γ_{t^{⋆}}} v̂; \frac{1}{λ − a_{inf}} inf_{Qt̄^{⋆}} (e^{−λt}(Lv)(x, t))\},  (17)

for any λ > a_{sup}. From (17) and from the definition of the function v̂ in (7), we obtain that if v|_{Γ_{t^{⋆}}} ≥ 0 and (Lv)|_{Qt̄^{⋆}} ≥ 0, then v(x, t^{⋆}) ≥ 0 on Qt̄^{⋆}. This completes the proof.

For simpler operators, similar statements can be found in [3, 9].

Theorem 3.4 If a_{0} ≤ 0, then operator (5) possesses both the weak maximum principle and the weak boundary maximum principle.

Proof. It is trivial that 1, t ∈ dom L, thus we can apply Theorem 2.12. The statement of the theorem follows from the facts that, under the conditions of the theorem, L1 = −a_{0} ≥ 0 and Lt = 1 − a_{0}t ≥ 1.

Theorem 3.5 If a_{0} = 0, then operator (5) possesses all the maximum principles.

Proof. We apply Theorem 2.13. The statement of the theorem follows from the facts that L1 = −a_{0} = 0 and Lt = 1 − a_{0}t = 1.

In order to analyse the necessity of the condition in Theorem 3.4, we introduce the following notation. The set of those real values ω for which the condition a_{0} ≤ ω implies the weak maximum principle for all operators in the form (5), will be denoted by M_{W}.

Theorem 3.6 The set M_{W} is identical with the set IR_{0}^{−}.

Proof. In view of Theorem 3.4, the inclusion IR_{0}^{−} ⊂ M_{W} is trivial. In order to show the inclusion in the opposite direction, let us choose an arbitrary positive constant γ and consider the one-dimensional operator

L ≡ \frac{∂}{∂t} − \frac{γ}{2} \frac{∂^{2}}{∂x^{2}} − γ,  (18)

where the set Q_{T} is defined to be Q_{T} = (0, π) × (0, T). Clearly, this operator has the form (5). Moreover a_{0} = γ > 0.

Let us choose the function v(x, t) = (γ/2)e^{γt/2} sin x, for which function the relation Lv(x, t) = 0 is true. Thus, we have

sup_{Q̄_{t^{⋆}}} v = \frac{γ}{2} e^{γt^{⋆}/2} > \frac{γ}{2} = max\{0, sup_{Γ_{t^{⋆}}} v\}

for any t^{⋆} ∈ (0, T). This shows that the weak boundary maximum principle does not hold for (18). We note that, due to the implications in Theorem 2.10, any of the maximum principles cannot be valid. Thus, the weak maximum principle does not hold either.

We investigate the condition of Theorem 3.5. Let us consider the set of those non-positive real values μ for which the condition μ ≤ a_{0} ≤ 0 implies all the maximum principles for all operators in the form (5). This set will be denoted by M_{A}. The zero upper bound for the function a_{0} is justifiable by the previous theorem.

Theorem 3.7 The set M_{A} is identical with the set {0}.

Proof. Because of the relation μ ≤ a_{0} ≤ 0, the operator L possesses both the weak maximum principle and the weak boundary maximum principle by all the choices of μ (see Theorem 3.4). Thus, it is enough to investigate the validity of the strong boundary maximum principle.

In view of Theorem 3.5, the inclusion 0 ∈ M_{A} is trivial. Now let γ be an arbitrary negative number and consider the operator

L ≡ \frac{∂}{∂t} + \frac{γ}{2} \frac{∂^{2}}{∂x^{2}} − γ,  (19)

where the set Q_{T} is defined again to be Q_{T} = (0, π)×(0, T). Here a_{0} = γ < 0. We set v(x, t) = −\frac{γ}{2}e^{γt/2}(sin x − 2), for which

Lv(x, t) = \frac{γ^{2}}{2} e^{γt/2}(sin x − 1) ≤ 0.

With this function v, we get the relation

max_{Q̄_{t^{⋆}}} v = \frac{γ}{2} e^{γt^{⋆}/2} > max\{γe^{γt^{⋆}/2}, γ/2\} = max_{Γ_{t^{⋆}}} v

for any t^{⋆} ∈ (0, T). Thus, the strong boundary maximum principle is not satisfied. Theorem 2.10 Implication i) gives that the strong maximum principle cannot be valid either. This completes the proof.

Remark 3.8 The operator in the proof of Theorem 3.7 shows that the validity of the weak (resp. weak boundary) maximum principle does not imply the strong (resp. strong boundary) one. Namely, while the strong maximum principle and the strong boundary maximum principle break to hold for operator (19), the weak maximum principle and the weak boundary maximum principle are valid. Indeed,

sup_{Q̄_{t^{⋆}}} v = \frac{γ}{2} e^{γt^{⋆}/2} ≤ 0 = max\{0, \frac{γ}{2} , γe^{γt^{⋆}/2}\} =

= max{0, sup_{Γ_{t^{⋆}}} v} + t^{⋆} max{0, sup_{Qt̄^{⋆}} Lv}

and

sup_{Q̄_{t^{⋆}}} v = \frac{γ}{2} e^{γt^{⋆}/2} ≤ 0 = max\{0, \frac{γ}{2} , γe^{γt^{⋆}/2}\} = max\{0, sup_{Γ_{t^{⋆}}} v\}.

After analyzing the condition in Theorem 3.4, we investigate a special operator

L = \frac{∂}{∂t} − \frac{∂^{2}}{∂x^{2}} − γ,  (20)

which is widely used in applications: the so-called one-dimensional heat conduction operator with linear source term.

Theorem 3.9 The operator L in (20) does not posses the weak boundary maximum principle for all constant γ ≥ 1.

Proof. We set Q_{T} = (0, π) × (0, T) again, and t^{⋆} is an arbitrary value from the interval (0, T).

Let γ > 1. We consider the function

v(x, t) = e^{(γ−1)t} sin x,

for which trivially Lv = 0 in Q_{T}, and v(0, t) = v(π, t) = 0 for all t ∈ [0, T]. For this function we have

sup_{Q̄_{t^{⋆}}} v = e^{(γ−1)t^{⋆}} > 1 = max{0, sup_{Γ_{t^{⋆}}} v}.

Thus, the operator (20) does not possess the weak boundary maximum principle for the values γ > 1.

Let γ = 1, and we consider the function

v(x, t) = \frac{8}{π} ∑_{{k=1 \atop k is odd}}^{∞} \frac{1}{k^{3}} e^{(1−k^{2})t} sin(kx)

for which Lv = 0 in Q_{T}, and v(0, t) = v(π, t) = 0 for all t ∈ [0, T]. It is known from the theory of Fourier series that

v(x, 0) = \frac{8}{π} ∑_{{k=1 \atop k is odd}}^{∞} \frac{1}{k^{3}} sin(kx) = x(π − x),

that is

max_{x∈[0,π]}{v(x, 0)} = π^{2}/4.

Supposing that t^{⋆} > 1/10, we obtain that

v(π/2, t^{⋆}) = \frac{8}{π} + \frac{8}{π} e^{t^{⋆}} ∑_{{k=3 \atop k is odd}}^{∞} \frac{(−1)^{(k−1)/2}}{k^{3}} e^{−k^{2}t^{⋆}} ≥

≥ \frac{8}{π} − \frac{8}{π} e^{t^{⋆}} ∑_{{k=3 \atop k is odd}}^{∞} \frac{1}{k^{3}} e^{−k^{2}t^{⋆}} ≥ \frac{8}{π} − \frac{8}{27π} e^{−8t^{⋆}} ∑_{{k=3 \atop k is odd}}^{∞} e^{−8(k−3)t^{⋆}} =

= \frac{8}{π} − \frac{8}{27π} e^{−8t^{⋆}} ∑_{l=0}^{∞}(e^{−16t^{⋆}})^{l} = \frac{8}{π} − \frac{8}{27π} e^{−8t^{⋆}} \frac{1}{1 − e^{−16t^{⋆}}} ≥

≥ \frac{8}{π} − \frac{8}{27π} e^{−4/5} \frac{1}{1 − e^{−8/5}} = 2.4934.

This yields that

sup_{Q̄_{t^{⋆}}} v ≥ 2.4934 > 2.4674 = \frac{π^{2}}{4} = max\{0, sup_{Γ_{t^{⋆}}} v\}.

This shows that the weak boundary maximum principle does not hold for the operator (20) for the value γ = 1 either.

Remark 3.10 The operator given in Theorem 3.9 would serve as a good example in Theorem 3.6 for the case γ ≥ 1. However, we note that Theorem 3.6 does not imply directly Theorem 3.9 for arbitrary positive γ values.

4 Summary

In this paper, we analyzed the qualitative properties of second order linear parabolic partial differential operators. We showed that these operators are monotone. If the coefficient function of v is less or equal to zero, then the operator possesses both the weak boundary maximum principle and the weak maximum principle. If the coefficient function is zero, then the operator fulfills all the discussed maximum principles. We gave examples that show that the obtained conditions are not only sufficient but they are necessary, too. In view of the generality of Section 2, our results can be applied not only to the investigated parabolic operator but also to other linear operators.

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(Received September 3, 2007)