## Abstract

In this paper, we establish extended maximum principles for solutions to linear parabolic partial differential inequalities on unbounded domains, where the solutions satisfy a variety of growth/decay conditions on the unbounded domain. We establish a conditional maximum principle, which states that a solution *u* to a linear parabolic partial differential inequality satisfies a maximum principle whenever a suitable weight function can be exhibited. Our extended maximum principles are then established by exhibiting suitable weight functions and applying the conditional maximum principle. In addition, we include several specific examples, to highlight the importance of certain generic conditions, which are required in the statements of maximum principles of this type. Furthermore, we demonstrate how to obtain associated comparison theorems from our extended maximum principles.

## 1. Introduction

Maximum principles are primarily used in the study of initial-boundary value problems to obtain *a priori* bounds on solutions, comparison theorems and uniqueness results (for example, see the established texts [1,2]). A secondary application of maximum principles can be found in the qualitative study of solutions to initial-boundary problems; some recent trends and open problems can be found in the texts [3–5] as well as in numerous others.

In this paper, we consider maximum principles for linear parabolic operators on unbounded domains. Specifically, let *Ω*. Associated with *Ω*, we introduce
*T*,*X*>0, with closures *L* be an operator that acts on sufficiently smooth functions *i*,*j*,≤*n*) are prescribed functions on *D*_{T}. When the matrix *A*(*x*,*t*)=(*a*_{ij}(*x*,*t*)) is symmetric and positive semi-definite for each (*x*,*t*)∈*D*_{T}, then we refer to *L* as a linear parabolic operator. The primary purpose of this paper is to extend the relationship between allowable spatial growth/decay as *L*[*u*]≤0 on *D*_{T}) and the conditions on the coefficients of the linear parabolic operator *L*, for which a maximum principle holds on *T*>0) such that *u*∈*C*^{2,1}(*D*_{T}) and
*k*_{1},*k*_{2}>0. Additionally, we also refer to *T*>0) such that *u*∈*C*^{2,1}(*D*_{T}) and
*k*_{1},*k*_{2}>0. A secondary purpose of the paper is to highlight the importance of certain generic conditions on the linear parabolic operators *L*, for maximum principles to hold, via the provision of specific examples.

We first give a brief summary of the development of maximum principles (occasionally referred to as Phragmèn Lindelöf principles) for linear parabolic partial differential inequalities on unbounded domains [1,6] related to those established in this paper. In [7], a maximum principle for a linear parabolic partial differential inequality on an unbounded domain was obtained, which complemented the non-uniqueness result for the linear heat equation obtained in [8]. Specifically, this maximum principle was designed for linear parabolic partial differential inequalities to allow uniqueness to be established for classical solutions to the linear heat equation, under the weakest possible growth conditions as *α*,λ≥0), were extensively developed (in particular, see [1,9–15] and references therein). In the development of this body of work, considerations regarding the optimum conditions on the associated maximum principles are rare; it is typical for a maximum principle to be established, without any discussion regarding limitations to the extension of the maximum principle, beyond the limitations of the method of proof.

More recently, in [16–20], via an alternative approach to that adopted in this paper, uniqueness results for initial-boundary value problems for parabolic partial differential equations have been established, with growth conditions specified on the solutions as *t*→0^{+}. To obtain these results, additional regularity on the coefficients *a*_{ij}, *b*_{i} and *c* in the linear parabolic operator *L* must be imposed, which we do not require for the results obtained in this paper. We also note that maximum principles for operators which have an additional coefficient *u*_{t} in (1.1), have been considered in [21], and although we do not consider these operators here, the approach we use can be readily adapted to accommodate these operators.

The main achievements of this paper comprise a generalization of the maximum principles established in [13,15] (which subsumed the results in [1,7,9,10,12]) for solutions to linear parabolic partial differential inequalities on unbounded domains with growth conditions on the solutions as *b*_{i} in the linear operator *L*. In addition, we extend the maximum principles established in [13,15] for solutions to linear parabolic partial differential inequalities on unbounded domains with decay conditions on the solutions as

The structure of the paper is as follows. In §2, we establish a weak maximum principle for a linear parabolic operator on unbounded domains, which is an extension of the classical weak maximum principle [22] onto unbounded domains. From this weak maximum principle, we obtain a widely applicable conditional maximum principle, and in doing so, illustrate how to obtain maximum principles for linear parabolic operators on unbounded domains with varying growth/decay conditions as *ϕ*. We also provide a subtle example to illustrate the importance of the conditions under which these maximum principles are obtained. In §3, we establish new maximum principles which generalize and extend the maximum principles contained in [13,15] by relaxing the conditions on the first-order coefficients *b*_{i} in the linear parabolic operator *L*, and considering additional classes of solutions of type (1.3), which are at most decaying as *ϕ* that allow applications of the conditional maximum principle established in §2. We complete the section by providing a function which demonstrates that our relaxation on the first-order coefficient in the linear parabolic operator is in a sense optimal, in that, at most it can be relaxed by a logarithmic growth in the spatial variables. In §4, we demonstrate briefly how these maximum principles can be applied to obtain comparison theorems and uniqueness results for a class of semi-linear parabolic initial-boundary value problems.

## 2. The conditional maximum principle

Here, we establish a conditional maximum principle for linear parabolic operators on an unbounded domain. This is in the spirit of those available for elliptic operators on bounded domains [1, ch. 2, section 9] and for parabolic operators on unbounded domains [6, pp. 211–214]. This conditional result reduces the proof of a maximum principle for a specified linear parabolic operator *L* to establishing the existence of a suitable weight function *ϕ*. First, we have the following.

### Definition 2.1

A linear parabolic operator *L* (defined on *D*_{T}) is said to satisfy condition (*H*) on a set *E*⊆*D*_{T} when *E*.

Definition 2.1 is associated with the classical maximum principle for a linear parabolic operator on a compact domain [1, pp. 174–175]. We now review a well-established maximum principle that plays a crucial role in obtaining our conditional maximum principle (for a similar result, see [11, p. 43]).

### Lemma 2.2

*Suppose that the linear parabolic operator* *L* *satisfies condition (H) on* *E*=*D*_{T}. *Moreover, suppose that* *is continuous with* *u*∈*C*^{2,1}(*D*_{T}) *and*
*and*
*while* *u*≤0 *on* ∂*D*_{T}. *Then*, *u*≤0 *on*

### Proof.

It follows from condition (*H*) that there exists a constant *C*>0 such that
*w* is continuous on *w*∈*C*^{2,1}(*D*_{T}) and *w*≤0 on ∂*D*_{T}. Additionally, via (2.1) and (2.4), it follows that
*L*. Furthermore, it follows from (2.2) and (2.4) that
*w*≤0 on *w* is bounded and continuous on *A*(*x**,*t**)=(*a*_{ij}(*x**,*t**)) is symmetric and positive semi-definite, it follows that there exists an invertible linear coordinate change
*r*=1,…,*n*, being the eigenvalues of *A*(*x**,*t**). Thus, it follows from (2.10) and (2.11) that
*w*, then it follows that

From lemma 2.2, we can now establish a conditional maximum principle that can be used to obtain maximum principles for parabolic operators not necessarily satisfying condition (*H*). This maximum principle is conditional as its application relies on the construction of a suitable weight function. We note that a similar concept is introduced in [6, p. 213].

### Lemma 2.3

*Let* *be continuous with* *u*∈*C*^{2,1}(*D*_{T}) *and* *u*≤0 *on* ∂*D*_{T}. *In addition, let* *L* *be a linear parabolic operator with* *L*[*u*]≤0 *on* *D*_{T}. *Suppose there exists a continuous function* *such that* *ϕ*>0 *on* *with* *ϕ*∈*C*^{2,1}(*D*_{T}) *and*
*Then*, *u*≤0 *on*

### Proof.

First, we define the function *w* is continuous, *w*∈*C*^{2,1}(*D*_{T}), *w*≤0 on ∂*D*_{T} and
*w* satisfies
*H*) on *D*_{T}, via (2.15)–(2.17), an application of lemma 2.2 gives
*u*≤0 on

It follows that the establishment of a maximum principle for a specific function *L* is reduced to finding a function *u* being bounded above on *x*>0, and *u*≤0 on *Ω*×{0} is replaced by

Before we establish new generic maximum principles in the following section, we give an example to illustrate the importance of condition (2.2) in lemma 2.2. Specifically, we produce a function *L* for which all of the conditions in lemma 2.2 are satisfied except that condition (2.2) is marginally violated, and for which the conclusion of lemma 2.2 is false. To begin, let *Ω*=∅) and introduce *u* is continuous on *u*∈*C*^{2,1}(*D*_{1}), with
*x*,*t*)∈*D*_{1}. Furthermore,
*u* is bounded on *x*,*t*)∈*D*_{1}, and so (2.27) corresponds to the inequality (1.1) with
*L* with *t*∈[0,1]. Moreover,
*b*(*x*,*t*) as *t*→0^{+} in *D*_{1} and leads to the resulting failure of lemma 2.2.

## 3. Maximum principles

Here, we apply lemma 2.3 to recover and extend the maximum principles developed in [1,7,9,10,12–15] for linear parabolic operators *L*, whose coefficients are constrained by the growth conditions of the unbounded solutions. For *α*,λ≥0, we obtain maximum principles for successively smaller sets of functions *L* are dependent on the set of functions *L*. Moreover, for *α*<0 or λ<0, we establish maximum principles of a form which have not been considered in any of the above works. We are able to make these extensions, following the careful consideration of the conditions on the first-order coefficients *b*_{i} in theorems 3.5 and 3.4 are logarithmically sharp and algebraically sharp, respectively. To begin, we have the following.

### Definition 3.1

Let *μ*,*p*_{1},*p*_{2}>0, for which,
*L* is said to satisfy condition (*H*)′ with *μ* and *ψ*, when there exists constants *i*≤*n*,
*x*,*t*)∈*D*_{T}.

We next establish the existence of a suitable weight function

### Lemma 3.2

*Let* *L* *be a linear parabolic operator which satisfies condition* (*H*)′ *with* *μ* *and* *ψ*. *Additionally, for any* *k*>0, *let*
*where*
*Then, the continuous function* *given by*,
*ϕ*>0 on *with* *ϕ*∈*C*^{2,1}(*D*_{δ}), *and*

### Proof.

Because *A*(*x*,*t*)=(*a*_{ij}(*x*,*t*)) is symmetric and positive semi-definite for all (*x*,*t*)∈*D*_{T}, then
*s*=(1+|*x*|^{2}). Observe that *ϕ*∈*C*^{2,1}(*D*_{δ}) and
*x*,*t*)∈*D*_{δ}. Thus, we have
*x*,*t*)∈*D*_{δ}. It now follows from (3.8) and definition 3.1 that

We can now establish a generalization of the maximum principle presented in [15]. We have the following.

### Theorem 3.3

*Let* *be continuous, with u∈C*^{2,1}*(D*_{T}*) and u≤0 on ∂D*_{T}*. In addition, let L be a linear parabolic operator which satisfies condition (H)′ with μ and ψ, and such that L[u]≤0 on D*_{T}*. When there exists k>0 such that*
*then u≤0 on*

### Proof.

Suppose there exists *k*>0 such that condition (3.13) is satisfied. With this value of *k*>0, set *δ*>0 and *u*≤0 on *δ*=*T*, the proof is complete. If *δ*≠*T*, then
*δ*=*δ*′, and 0<*T*−(*N*+1)*δ*′≤*δ*′. We may then take a final step with *δ*=*T*−(*N*+1)*δ*′, and so we have *u*≤0 on *T*=*δ*′+*Nδ*′+(*T*−(*N*+1)*δ*′)). ■

Next, we establish generalizations of the maximum principles given in [13,14] for solutions to partial differential inequalities in *α*,λ≥0. We present these maximum principles in descending order, in that the sets

### Theorem 3.4

*Let* *be continuous with* *for* *. In addition, let L be a linear parabolic operator which, for A,B,C≥0 satisfies*
*for all (x,t)∈D*_{T} *and 1≤i≤n. When u≤0 on ∂D*_{T} *and L[u]≤0 on D*_{T}*, then u≤0 on*

### Proof.

Let *ψ*(*η*)≥1 and
*μ*=1, satisfies conditions (3.1) and (3.2) in definition 3.1. From (3.16) and (3.17), we have
*L* satisfies condition (*H*)′ with *μ*=1 and *ψ* given by (3.15). Now, with *k*>0 such that
*L* satisfies condition (*H*)′ with *μ*=1 and *ψ* given by (3.15), an application of theorem 3.3, with (3.18), establishes that *u*≤0 on

### Theorem 3.5

*Let* *be continuous with* *for α=0,* *. In addition, let L be a linear parabolic operator which, for A,B,C≥0 satisfies*
*for all (x,t)∈D*_{T} *and 1≤i≤n. When u≤0 on ∂D*_{T} *and L[u]≤0 on D*_{T}*, then u≤0 on*

### Proof.

Let *L* satisfies condition (*H*)′ with *μ*=λ/(λ−1) and *ψ* given by (3.19). The remainder of the proof follows that of theorem 3.4. ■

Theorems 3.4 and 3.5 recover and extend the maximum principles, which have been developed chronologically in [7,9–12], and extend the maximum principles in [13,14]. We note that maximum principles are considered in [13], which have growth conditions which we have not considered here for the sake of brevity (these are obtained directly from lemma 2.3 with an appropriate weight function *ϕ*). We now focus our attention on the classes of solutions that decay as *a priori* defining the decay of the solution as

### Definition 3.6

Let *μ*>0 and the linear parabolic operator *L* be as in definition 3.1, with (3.1) replaced by
*x*,*t*)∈*D*_{T} and 1≤*i*≤*n*. When conditions (3.20), (3.2), (3.3), (3.21) and (3.5) are satisfied, then the linear parabolic operator *L* is said to satisfy condition (*H*)′′ with *μ* and *ψ*.

We now have the following.

### Lemma 3.7

*Let* *L* *be a linear parabolic operator which satisfies condition* (*H*)′′ *with* *μ* *and* *ψ*. *Additionally, for any* *k*<0, *let*
*where*
*Then, the continuous function* *given by*,
*satisfies* *ϕ*>0 *on* *with* *ϕ*∈*C*^{2,1}(*D*_{δ}), *and*

### Proof.

We proceed as in the proof of lemma 3.2 with *k*<0. It then follows that
*x*,*t*)∈*D*_{δ}. Now, it follows from (3.8) and definition 3.6 that

We now make a further extension of the maximum principle contained in [15] for solutions that satisfy a specified decay condition as

### Theorem 3.8

*Let* *be continuous, u∈C*^{2,1}*(D*_{T}*) and u≤0 on ∂D*_{T}*. In addition, let L be a linear parabolic operator which satisfies condition (H)′′ with μ and ψ, and such that L[u]≤0 on D*_{T}*. When there exists k<0 such that*
*then u≤0 on*

### Proof.

The proof follows the same steps as the proof of theorem 3.3. ■

We are now in a position to establish new maximum principles, of the type considered in [13,14] for solutions which satisfy specified decay conditions as *b*_{i}. We now have the following.

### Theorem 3.9

*Let* *be continuous with* *for α=0,* *. In addition, let L be a linear parabolic operator which, for A,B,C≥0 satisfies*
*for all (x,t)∈D*_{T} *and 1≤i≤n. When u≤0 on ∂D*_{T} *and L[u]≤0 on D*_{T}*, then u≤0 on*

### Proof.

For λ<−1, let *μ*=|λ|/(|λ|−1). It follows that *ψ*(*η*)≥1 and
*μ*=|λ|/(|λ|−1). It follows from (3.29), (3.30) and (3.28) that
*L* satisfies condition (*H*)′′ with *μ*=|λ|/(|λ|−1) and *ψ* given by (3.28). Furthermore, because *k*<0 such that

Complementary to this, we also have the following.

### Theorem 3.10

*Let* *be continuous with* *for* *. In addition, let L be a linear parabolic operator which, for A,B,C≥0 satisfies*
*for all (x,t)∈D*_{T} *and 1≤i≤n. When u≤0 on ∂D*_{T} *and L[u]≤0 on D*_{T}*, then u≤0 on*

### Proof.

Let *L* satisfies condition (*H*)′′ with *μ*=1 and *ψ* given by (3.31). The remainder of the proof follows that of theorem 3.9. ■

It is worth remarking that in [7,9–14,21,22], maximum principles are obtained where the condition on the first-order coefficient *L*, is bounded in modulus, namely for *B*≥0,

We now provide an example that illustrates the optimality of our condition on the first-order term *γ*>0 is constant. Observe that *w* is continuous on *w*∈*C*^{2,1}(*D*_{1}), where *α*,λ≥0). In addition,
*L*[⋅] is a linear parabolic operator of the form (1.1), with

### Remark 3.11

Observe that it is the growth rate of *x*→0 that leads to the resulting failure of theorem 3.5 (and theorem 3.4). Moreover, it follows that the condition on

It should also be noted that if a function *k*,*γ*>0), then theorem 3.5 implies that *u*≤0 on *xb*(*x*,*t*) in theorem 3.5 only requires the growth rate as *xb*(*x*,*t*)| as

To contextualize the nature of theorem 3.10 as an extension of the maximum principles in [15], it is illustrative to consider the following example. Let *β*∈(0,1]. For *L* given by (3.44), theorems 3.4 and 3.5 cannot be applied, owing to the unspecified growth of *c* is not bounded above on *D*_{1}. However, it follows that *L* given by (3.44) satisfies the conditions of theorem 3.10 with *α*=−*β* and λ=0, and hence, if *L*[*u*]≤0 with *u*≤0 on ∂*D*_{1}, then *u*≤0 on *L* given by (3.44) but with *β*>1, then *L* would not satisfy theorem 3.10, owing to the constant coefficient of the second-order term together with the growth of the coefficient of the zeroth-order term. Conversely, if we consider *L* given by (3.44) with *L* would satisfy the conditions of theorem 3.4 with *α*=*β* and λ=0, and hence, if *L*[*u*]≤0 with *u*≤0 on ∂*D*_{1}, then *u*≤0 on

## 4. Applications

Here, we demonstrate how the maximum principles, we have developed in §3, can be used to establish comparison theorems. These comparison theorems can then be used to establish uniqueness results for the following semi-linear parabolic initial-boundary value problem, which commonly arises in both applied and theoretical studies of partial differential equations (see, for example, the recent texts [3–5], and the classical texts [11,6]). We restrict attention to bounded solutions (that is, in *u*∈*C*^{2,1}(*D*_{T}), such that
*L* is a linear parabolic operator as in (1.1), and
*u*∈*C*^{2,1}(*D*_{T}), and which satisfies (4.1) and (4.3) is referred to as a solution of the initial-boundary value problem (IBVP) with linear parabolic operator *L*, nonlinearity *f* and initial-boundary data *g*. Before we establish our results relating to (IBVP), we require two definitions.

### Definition 4.1

Let *L* is a linear parabolic operator and, *regular subsolution* and *regular supersolution* to (IBVP) with linear parabolic operator *L*, nonlinearity *f* and initial-boundary data *g*.

### Definition 4.2

The function *H*)_{α} with *α*≥0 when for any closed bounded interval *k*_{M}>0 such that for all *u*,*v*∈*M* with *u*≥*v*, *f* satisfies the inequality

The following observation is useful.

### Remark 4.3

Let *f* satisfy condition (*H*)_{α} with *α*≥0, then on every closed bounded interval *k*_{M}>0 such that for all *u*,*v*∈*M* with *u*≠*v*, then,
*f* is locally Lipschitz continuous in *u*, uniformly on *D*_{T}, namely for all *u*,*v*∈*M*, there exists a constant *k*_{M}>0 such that
*f* satisfies condition (*H*)_{α} for all *α*≥0.

We now establish the following comparison theorem for (IBVP).

### Theorem 4.4

*Let* *and* *be a regular supersolution and a regular subsolution to (IBVP) with linear parabolic operator L, nonlinearity f and initial-boundary data g, respectively. Moreover, suppose that for some α≥0, f satisfies condition (H)*_{α}*, and there exists constants A,B,C≥0 such that the coefficients of the linear parabolic operator L satisfy*
*for all (x,t)∈D*_{T} *and 1≤i≤n. Then,*

### Proof.

Define *w*(*x*,*t*)∈*M* for all *D*_{T}, we have via definition 4.1,
*L*, and
*k*_{M}>0 such that
*α*>0 or theorem 3.5 when *α*=0. Moreover, via definition 4.1,
*α*>0) or theorem 3.5 (*α*=0), with (4.5) and (4.6), establishes that

We are now able to establish uniqueness of solutions to IBVP.

### Theorem 4.5

*Suppose that* *satisfies condition (H)*_{α} *for some α≥0, and there exists constants A,B,C≥0 such that the coefficients of the linear parabolic operator L satisfy*
*for all (x,t)∈D*_{T} *and 1≤i≤n. Then, (IBVP) with linear parabolic operator L, nonlinearity f and initial-boundary data g has at most one solution on*

### Proof.

Let *L*, nonlinearity *f* and initial-boundary data *g* on *u* is a solution to IBVP with linear parabolic operator *L*, nonlinearity *f* and initial-boundary data *g* on *u* is both a regular supersolution and a regular subsolution to IBVP with linear parabolic operator *L*, nonlinearity *f* and initial-boundary data *g* on *u*^{(1)} and *u*^{(2)} to be a regular subsolution and a regular supersolution to IBVP with linear parabolic operator *L*, nonlinearity *f* and initial-boundary data *g*, respectively, then via theorem 4.4 we have,
*u*^{(1)}=*u*^{(2)} on

## Acknowledgements

J.C.M acknowledges financial support from the EPSRC.

- Received January 30, 2014.
- Accepted April 2, 2014.

© 2014 The Authors. Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0/, which permits unrestricted use, provided the original author and source are credited.