Extended weak maximum principles for parabolic partial differential inequalities on unbounded domains

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.

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.

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 initialboundary problems; some recent trends and open problems can be found in the texts [3][4][5] as well as in numerous others.
In this paper, we consider maximum principles for linear parabolic operators on unbounded domains. for T, X > 0, with closuresD T andD X T . Here,D T = D T ∪ ∂D T . In addition, let L be an operator that acts on sufficiently smooth functions u : D T → R, given by where a ij , b i , c : D T → R (1 ≤ 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 |x| → ∞, of solutions to linear parabolic partial differential inequalities (L[u] ≤ 0 on D T ) and the conditions on the coefficients of the linear parabolic operator L, for which a maximum principle holds onD T . In this respect, it is convenient to introduce E λ α , for α ∈ [0, ∞), λ ∈ [0, ∞) as the set of continuous functions u :D T → R (some T > 0) such that u ∈ C 2,1 (D T ) and u(x, t) ≤ k 1 e k 2 (1+|x| 2 ) α (1+ln (1+|x| 2 )) λ ∀ (x, t) ∈D T (1.2) for some k 1 , k 2 > 0. Additionally, we also refer to E λ α , for α ∈ (−∞, 0], λ ∈ (−∞, 0] as the set of continuous functions u :D T → R (some T > 0) such that u ∈ C 2,1 (D T ) and u(x, t) ≤ k 1 e −k 2 (1+|x| 2 ) |α| (1+ln (1+|x| 2 )) |λ| ∀ (x, t) ∈D T (1.3) for some 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 |x| → ∞. Following these works, maximum principles for linear parabolic partial differential inequalities on unbounded domains, with specific growth conditions on the solutions as |x| → ∞, which have the general form given in (1.2) (for various values of α, λ ≥ 0), were extensively developed (in particular, see [1,[9][10][11][12][13][14][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][17][18][19][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 |x| → ∞ and 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 d : D T → R, multiplying the term 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 |x| → ∞ of the form given in (1.2), which we henceforth refer to as type (1.2) growth conditions. We achieve this via a relaxation of the conditions in [13,15], on the coefficients 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 |x| → ∞ of the form given in (1.3), which we henceforth refer to as type (1.3) decay conditions, which have not been not considered in any of the previously mentioned works. We highlight this because, in numerous applications of maximum principles, the rate of decay of the solution as |x| → ∞ to the parabolic partial differential inequality is known, and hence, our results may be applicable, whereas the maximum principles designed for solutions with growth conditions as |x| → ∞ of type (1.2) may be inapplicable. Additionally, we have constructed specific examples to highlight that extensions to these maximum principles, in certain generic directions, are not possible.
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 |x| → ∞. This maximum principle is conditional because it depends on the existence of a suitable weight function 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 |x| → ∞. We achieve this by establishing the existence of suitable weight functions 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 firstorder 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.

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.
Proof. It follows from condition (H) that there exists a constant C > 0 such that Now, we define the function w :D T → R given by It follows immediately that w is continuous onD T , w ∈ C 2,1 (D T ) and w ≤ 0 on ∂D T . Additionally, via (2.1) and (2.4), it follows that We now show that for any n ∈ N and hence that w ≤ 0 onD T . Suppose that (2.8) is false. Then, via (2.7), because w is bounded and continuous onD X n T , it follows that there exists (x * , t * ) ∈ D X n T such that Then, via (2.9), (2.5) and (2.3), it follows that Now, because the matrix A(x * , t * ) = (a ij (x * , t * )) is symmetric and positive semi-definite, it follows that there exists an invertible linear coordinate change λ * r w y r y r (2.11) with λ * r ≥ 0, r = 1, . . . , n, being the eigenvalues of A(x * , t * ). Thus, it follows from (2.10) and (2.11) that Now, because (x * , t * ) ∈ D X n T is a local maxima of w, then it follows that w t (x * , t * ) ≥ 0 and w y i y i (x * , t * ) ≤ 0, (2.13) and so, for each n ∈ N, and so 14) The result follows from (2.4) and (2.14).
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].
Proof. First, we define the function w : Moreover, we observe that w satisfies Because the linear parabolic operatorL satisfies condition (H) on D T , via (2.15)-(2.17), an application of lemma 2.2 gives w ≤ 0 onD T and hence, via (2.15), that u ≤ 0 onD T , as required.
It follows that the establishment of a maximum principle for a specific function u :D T → R and a specific linear parabolic operator L is reduced to finding a function φ :D T → R which satisfies the conditions in lemma 2.3. An advantage of this conditional maximum principle is that not only can it be used to develop generic maximum principles, as we demonstrate in §3, but it can also be used to obtain maximum principles for specific problems which do not adhere to the conditions of the available generic maximum principles, if a suitable weight function φ :D T → R can be found. We also note that, without further technical difficulties, lemma 2.3, with suitable minor modifications in statement, can be established when u : 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 u :D T → R and a linear parabolic operator 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 = R (and so ∂Ω = ∅) and introduce u : It is readily established that u is continuous onD 1 and for all (x, t) ∈D 1 and so u is bounded onD 1 . Additionally, for all (x, t) ∈ D 1 , and so (2.27) corresponds to the inequality (1.1) with  (2.2), and for which the conclusion of lemma 2.2 is violated, via (2.25). We now consider how this example violates condition (2.2). We observe from (2.18) that However, this limit is not uniform for t ∈ [0, 1]. Moreover, . This feature is related to the unboundedness of b(x, t) as t → 0 + in D 1 and leads to the resulting failure of lemma 2.2.

Maximum principles
Here, we apply lemma 2.3 to recover and extend the maximum principles developed in [1,7,9,10,[12][13][14][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 E λ α (as in (1.2)) where the conditions on the coefficients of the linear parabolic operators L are dependent on the set of functions E λ α . In particular, we establish a generalization of the maximum principle in [15] (which itself, recovered and generalized the results in [1,7,9,10,[12][13][14]), via a relaxation of the condition on the first-order coefficients in the linear parabolic operator L. Moreover, for E λ α , as in (1.3), with 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 : D T → R in the linear parabolic operator. At the end of this section, we give examples of functions which exhibit that the conditions under which the following maximum principles are established are, in some sense, optimal, namely that the conditions on 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 ψ ∈ C 2 ([1, ∞)) be a positive strictly increasing function such that there exist constants μ, p 1 , p 2 > 0, for which, for all η ∈ [1, ∞). A linear parabolic operator L is said to satisfy condition (H) with μ and ψ, when there exists constantsĀ,B,C ≥ 0 such that for 1 ≤ i ≤ n, and c(x, t) ≤C(ψ(1 + |x| 2 )) μ , (3.5) for all (x, t) ∈ D T .
We next establish the existence of a suitable weight function φ :D T → R which may be used in applications of lemma 2.3. In the following result, we provide an extension, for our purpose, of that in [15, lemma 2].
We can now establish a generalization of the maximum principle presented in [15]. We have the following.
Next, we establish generalizations of the maximum principles given in [13,14] for solutions to partial differential inequalities in E λ α with α, λ ≥ 0. We present these maximum principles in descending order, in that the sets E λ α in the following theorems get subsequently smaller while the conditions on the coefficients in the linear parabolic operator relax, tighten and switch sign (see theorems 3.4, 3.5, 3.9 and 3.10). Theorem 3.4. Let u :D T → R be continuous with u ∈ E λ α for α ∈ (0, ∞), λ ∈ [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 onD T .
Theorem 3.5. Let u :D T → R be continuous with u ∈ E λ α for α = 0, λ ∈ (1, ∞). In addition, let L be a linear parabolic operator which, for A, B, C ≥ 0 satisfies It is readily verified that 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][10][11][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 |x| → ∞, which have received much less attention in the literature. Generally, when considering solutions in this class, results with similar coefficient conditions to those in lemma 2.2, theorems 3.4 or 3.5 are applied to obtain maximum principles; however, these can be considerably improved by a priori defining the decay of the solution as |x| → ∞. To begin, we require the following. 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 Then, the continuous function φ :D δ → R, given by, Proof. We proceed as in the proof of lemma 3.2 with φ :D δ → R given by (3.23), and k < 0. It then follows that for all (x, t) ∈ D δ . Now, it follows from (3.8) and definition 3.6 that In addition, via definition 3.6, we have Therefore, it follows via (3.24)-(3.27) that as required.
We now make a further extension of the maximum principle contained in [15] for solutions that satisfy a specified decay condition as |x| → ∞.  e k(ψ(1+|x| 2 )) μ ≤ 0, Proof. The proof follows the same steps as the proof of theorem 3.3.
Complementary to this, we also have the following.
Proof. Let ψ : [1, ∞) → R be given by It is readily verified that 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][10][11][12][13][14]21,22], maximum principles are obtained where the condition on the first-order coefficient b i : D T → R in the linear operator L, is bounded in modulus, This contrasts to the one-sided bounds in the statements of the maximum principles obtained here. We now provide an example that illustrates the optimality of our condition on the first-order term b i : D T → R in both theorems 3.4 and 3.5. With Ω = R, we consider w :D 1 → R given by, where γ > 0 is constant. Observe that w is continuous onD 1 and w ∈ C 2,1 (D 1 ), where w t : D 1 → R and w x : D 1 → R are given by, and (3.34) and so w ∈ E λ 0 for all λ ≥ 0 (and hence w ∈ E λ α for all α, λ ≥ 0). In addition,  Remark 3.11. Observe that it is the growth rate of b : D 1 → R given by (3.39) as |x| → ∞, and not the behaviour as x → 0 that leads to the resulting failure of theorem 3.5 (and theorem 3.4). Moreover, it follows that the condition on b i : D T → R in theorem 3.5 is logarithmically sharp, namely the condition on b i : D T → R cannot be relaxed to allow larger logarithmic growth as |x| → ∞, without altering other conditions. Additionally, it follows that the condition on b i : D T → R in theorem 3.4 is algebraically sharp, namely the condition on b i : D T → R cannot be relaxed to allow larger algebraic growth as |x| → ∞, without altering other conditions. However, additional logarithmic growth, as in the conditions of theorem 3.5, is perhaps possible.
It should also be noted that if a function u :D 1 → R satisfies the conditions of theorem 3.5, with coefficients a, b, c : D 1 → R given by (3.40) b(x, t) = −k(log (1 + x 2 )) γ x 3 ∀ (x, t) ∈ D 1 , (3.43) and (3.41) respectively (with constants k, γ > 0), then theorem 3.5 implies that u ≤ 0 onD 1 , despite the superlinear growth of b : D 1 → R as |x| → ∞, given by (3.43), because the inequality on xb(x, t) in theorem 3.5 only requires the growth rate as |x| → ∞ to be limited from above. Such cases would be precluded in the maximum principles in [15], which require growth rate limitations on |xb(x, t)| as |x| → ∞. We also note that in [10, p. 17], an example is given that violates the conclusion of theorem 3.5; however, in this example, the conditions on both a : D 1 → R and b : D 1 → R are violated, and hence, it is more difficult to draw conclusions from it.
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 Ω = R and introduce the linear parabolic operator withb : R → R an increasing function without growth limitations as |x| → ∞, and β ∈ (0, 1]. For L given by (3.44), theorems 3.4 and 3.5 cannot be applied, owing to the unspecified growth ofb as |x| → ∞. Moreover, lemma 2.2 cannot be applied since 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 u ∈ E 0 −β satisfies L[u] ≤ 0 with u ≤ 0 on ∂D 1 , then u ≤ 0 onD 1 . In addition, note that if we consider 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) withb : R → R being a decreasing function, then L would satisfy the conditions of theorem 3.4 with α = β and λ = 0, and hence, if u ∈ E 0 β satisfies L[u] ≤ 0 with u ≤ 0 on ∂D 1 , then u ≤ 0 onD 1 .

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][4][5], and the classical texts [6,11]). We restrict attention to bounded solutions (that is, in E 0 0 ) of initial-boundary value problems for semi-linear parabolic equations, for brevity, with results for unbounded/decaying solutions following similarly. Additionally, we note that comparison theorems and uniqueness results can be established for bounded solutions to initial-boundary value problems for quasi-linear/nonlinear parabolic equations via a similar approach to that which follows, provided appropriate restrictions on the quasi-linear/nonlinear terms hold (see, for example, [1] or [6]). Now, let u :D T → R be continuous and bounded, and u ∈ C 2,1 (D T ), such that where L is a linear parabolic operator as in (1.1), and is a prescribed function, while where g : ∂D T → R is a given function, which is bounded and continuous. A continuous and bounded function u :D T → R with 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. Letū, u :D T → R be continuous and bounded, andū, u ∈ C 2,1 (D T ). Suppose further that where L is a linear parabolic operator and, f : D T × R → R and g : ∂D T → R are prescribed functions. Then, onD T , u is called a regular subsolution andū is called a regular supersolution to (IBVP) with linear parabolic operator L, nonlinearity f and initial-boundary data g.
The following observation is useful.
Furthermore, it follows that if 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 then f satisfies condition (H) α for all α ≥ 0.
We are now able to establish uniqueness of solutions to IBVP.  c(x, t) ≤ C(1 + |x| 2 ) α for all (x, t) ∈ D T and 1 ≤ i ≤ n. Then, (IBVP) with linear parabolic operator L, nonlinearity f and initialboundary data g has at most one solution onD T .
Proof. Let u (1) :D T → R and u (2) :D T → R both be solutions to (IBVP) with linear parabolic operator L, nonlinearity f and initial-boundary data g onD T . It is trivial to show that if u is a solution to IBVP with linear parabolic operator L, nonlinearity f and initial-boundary data g on D T then, via Definition 4.1, u is both a regular supersolution and a regular subsolution to IBVP with linear parabolic operator L, nonlinearity f and initial-boundary data g onD T . On taking 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, (4.7) A symmetrical argument establishes that u (2) ≤ u (1) onD T , (4.8) and therefore, via (4.7) and (4.8), it follows that u (1) = u (2) onD T , as required.