1 Evolution equations and positivity
To set the stage, we start with a reminder about linear evolution equations whose solutions are positive whenever the initial value is.
Linear ODEs and positivity
For a matrix , the linear and autonomous initial value problem
where , is well-known to be solved by the function
We say that the matrix family is positive if for all whenever ; equivalently, for all . Here, we use the notation for a vector or a matrix to say that all its entries are .
Remark 1. There is some terminological inconsistency in the literature with respect to this notion: in matrix analysis and in some parts of PDE theory, it is common to use the word non-negativity; we use the notion positivity instead, which is more common in functional analysis.
To get an intuition for this concept, it is useful to recall that positivity of the matrix exponential function is easy to characterise in terms of .
For , the family is positive if and only if every off-diagonal entry of is .
Proof. “” For indices , one has
where are the canonical unit vectors and denotes the standard inner product on .
“” By assumption, one has, for a sufficiently large number , the inequality , and hence
for all , where the inequality at the end follows from the series expansion of the matrix exponential function. ∎
A typical situation where positivity of matrix exponential functions occurs is the study of Markov processes on finite state spaces.
Example 3. Assume that all off-diagonal entries of are and that all rows of sum up to . Then is positive, and the vector whose entries are all equal to satisfies and thus for all . This shows that each of the matrices , , is row stochastic, so describes a continuous-time Markov process on the finite state space .
In infinite dimensions, we are still interested in initial value problems of the form
but this time, is an element of a Banach space , and is a linear operator which is defined on a vector subspace of . The initial value problem is well-posed if and only if is a generator of a -semigroup. Such a -semigroup is a family of bounded linear operators on which is a suitable infinite-dimensional substitute of the matrix exponential function and has similar properties, but it is not given by an exponential series in general. The solution to the initial value problem is then given, again, by the formula for . The generator and the -semigroup determine each other uniquely, and the relation between semigroup and generator can in general be expressed by the formula
The following notion will be used several times later on. For a linear operator on a Banach space , the quantity
where denotes the spectrum of , is called the spectral bound of . If generates a -semigroup, then (see [20, Theorem II.1.10 (ii)]). More information about -semigroup theory can be found, for instance, in the monographs [31, 20].
Let us briefly illustrate the concept of a -semigroup by two very classical examples.
Examples 4. (a) Let and let the operator be the Laplace operator on the space , i.e.
Then generates a -semigroup on that is given by the formula
for and . The semigroup is called the heat semigroup since it describes the solutions to the heat equation
Similar observations can be made on the space , but the domain of the Laplace operator cannot be chosen to be a Sobolev space in that case, due to the lack of elliptic regularity.
(b) Let and let the operator be the negative first derivative on , given by
Then generates the so-called right shift semigroup on given by
for . The mapping is a so-called mild solution to the transport equation
see [20, Definition II.6.3] for the definition of mild solutions. This example is an easy illustration of the general principle that boundary conditions of a PDE are encoded in the domain of the corresponding operator .
In order to discuss positive-semigroups, one needs an order structure on the underlying Banach space . This can be for instance a partial order induced by a general closed convex cone, or more specifically the order structure of a Banach lattice. To facilitate the exposition here, we will restrict our attention to the illustrative case of function spaces, most importantly to -spaces (over -finite measure spaces).
For a function , we write to indicate that for almost all . In accordance with the terminology used above, we call a function positive if it satisfies . A -semigroup on is called positive if for all whenever . Equivalently, each of the operators is positive – which we denote by – in the sense that it maps positive functions to positive functions.
We have already encountered two examples of positive -semigroups: as is easy to see, both semigroups in Examples 4 are positive.
2 Positivity for large times
Let us now proceed to a more surprising situation, where positive initial values lead to solutions which might change sign at first, but again become – and stay – positive for sufficiently large times. In this section, we illustrate by means of two easy examples that this kind of behaviour can indeed occur; a more systematic account is presented in the subsequent section.
A matrix example
Let us start with a simple three-dimensional example.
Example 5. Consider the orthonormal basis of consisting of the three vectors
Let be such that its representation matrix with respect to the basis is given by
i.e., we let , where consists of the columns . A direct computation shows that has some strictly negative off-diagonal entries, so is not positive according to Theorem 2. On the other hand, has the eigenvalue (with eigenvector ) as well as the further eigenvalues , so converges to the matrix , whose entries are all equal to , as ; this shows that is a positive matrix for all sufficiently large times .
A fourth order PDE
Let us now discuss an infinite-dimensional example where eventual positivity occurs.
Example 6. Let us consider the biharmonic heat equation with periodic boundary conditions on . It is given by
where has domain
and is given by for each . The -semigroup is not positive; this can for instance be seen by associating a sesquilinear form to and using the so-called Beurling–Deny criterion [30, Corollary 2.18].
However, we can prove positivity for large times. To this end, note that the operator is self-adjoint, and its spectrum consists of isolated eigenvalues only since embeds compactly into . The largest eigenvalue of is , and the constant function spans the corresponding eigenspace. Hence we conclude, for instance from the spectral theorem for self-adjoint operators with compact resolvent, that
for each as . Since is self-adjoint, the operators have the property that for they map into and thus into . Moreover, they are even continuous from to (this follows for instance from the closed graph theorem), so for , we even have
as , where the convergence takes place with respect to the norm in . This implies that if , then for all sufficiently large times .
3 A systematic theory
After the previous ad hoc examples, we now present a few excerpts of a more systematic account of eventual positivity.
Eventually positive matrix semigroups
Example 5 already gives quite a straightforward idea of how to obtain a sufficient condition for a matrix exponential function to be eventually positive: if a matrix has a simple real eigenvalue that dominates the real parts of all other eigenvalues and if the corresponding eigenvectors of and the transposed matrix have strictly positive entries only, then we expect to be positive – and in fact to even have strictly positive entries only – for all sufficiently large . A bit more surprising is the Perron–Frobenius-like fact that the converse implication also holds. This was proved by Noutsos and Tsatsomeros in [29, Theorem 3.3], who thus obtained the following theorem (in a slightly different form; see [17, Theorem 6.1] for the following version of the result).
For a matrix , the following assertions are equivalent.
There exists a time such that all entries of are strictly positive for all .
The spectral bound is a geometrically simple eigenvalue of and strictly larger than the real part of every other eigenvalue of . Moreover, both and have a strictly positive eigenvector for , respectively.
Here, a strictly positive vector means a vector whose entries are all strictly positive.
Individual vs. uniform behaviour
In infinite dimensions, there is a subtlety that we have not properly discussed yet. Let be a -semigroup on a function space . If, for every , there exists a time such that for all , it is natural to call the semigroup individually eventually positive since might depend on . If in addition can be chosen to be independent of , then we call the semigroup uniformly eventually positive.
In finite dimensions, the two concepts can be easily seen to coincide (just apply the semigroup to all canonical unit vectors), but in infinite dimensions, there exist semigroups which are individually but not uniformly eventually positive [17, Examples 5.7 and 5.8].
Conditions for eventual positivity in infinite dimensions
The arguments given in Example 6 show individual eventual positivity of the semigroup, and the same argument can easily be generalised to a more abstract setting. There is one important issue to note, though: if the leading eigenfunction is not bounded away from , but might be equal to on the boundary of the underlying domain (as in the case of Dirichlet boundary conditions), then it no longer suffices for the argument that be contained in ; instead, one needs the condition that every vector in is dominated by a multiple of the leading eigenfunction. This property is closely related to Sobolev embedding theorems, and can be used to give a characterisation of a certain strong version of individual eventual positivity that is reminiscent of Theorem 7.
On the other hand, giving conditions for uniform rather than individual eventual positivity is more subtle. It requires a domination condition not only on the vectors in the image of , but also on the image of the dual operator. If the semigroup is self-adjoint, though, this dual condition becomes redundant and one ends up with the following sufficient condition for uniform eventual positivity.
Let be a -finite measure space, let be a self-adjoint -semigroup on which leaves the set of real-valued functions invariant, and let be a function which is strictly positive almost everywhere. Assume that the following assumptions hold.
The spectral bound is a simple eigenvalue of , and the corresponding eigenspace contains a function satisfying for a number .
There exists a time such that the modulus of every vector in is dominated by a multiple of .
Then is uniformly eventually positive.
The really interesting part in the conclusion of the theorem is the word uniformly, and this is more involved than the argument presented in Example 6. Two different proofs of the theorem are known: the first one is based on an eigenvalue estimate and the theory of Hilbert–Schmidt operators [24, Theorem 10.2.1] (the assumptions in the reference are slightly different, but the same argument works under the assumptions presented above); the second one employs a duality argument and can thus be generalised to non-self-adjoint semigroups on more general spaces [14, Theorem 3.3 and Corollary 3.5]. This reference also shows that the theorem can be adjusted to even yield a characterisation of a stronger type of eventual positivity.
Positive semigroups are known to have surprising structural properties, in particular with regard to their spectrum. For some of these properties, it can be shown that they are shared by eventually positive semigroups, though some of the proofs are different from the classical proofs for the positive case. Here are two examples.
If the spectrum of the generator of an individually eventually positive semigroup is non-empty, then it follows that the spectral bound is itself a spectral value [17, Theorem 7.6].
For uniformly eventually positive semigroups on -spaces, the spectral bound coincides with the so-called growth bound of the semigroup (see e.g. [20, Definition I.5.6] for a definition); this was recently shown by Vogt [35, Theorem 2]. The same can be shown, even for individually eventually positive semigroups, on spaces of continuous functions [6, Theorem 4].
More results on the spectrum of eventually positive -semigroups can be found in .
4 More examples
The biharmonic heat equation
Example 6 can be adjusted in the following way: we replace the unit interval with a ball in , the fourth derivative with the square of the Laplace operator, and the periodic boundary conditions with so-called clamped plate boundary conditions, which require both the function and its normal derivative to vanish at the boundary. On , this yields the operator given by
where and denote Sobolev spaces. The operator is self-adjoint and has negative spectral bound. It thus generates a -semigroup which describes the solutions to the so-called bi-harmonic heat equation
We have the following result.
The bi-harmonic heat semigroup on is uniformly eventually positive.
Rough outline of the proof. Since is a ball, the inverse operator – or rather its integral kernel, the so-called Green function of – can be computed explicitly, and this was in fact done by Boggio over a hundred years ago  (see also [27, Section 2]). The explicit formula shows that maps positive functions to positive functions, and even strengthens their positivity in an appropriate sense. Hence, by a Krein–Rutman type result, we obtain that the leading eigenfunction of is strictly positive inside . Given the specific boundary conditions, it is not too surprising that we also get that assumptions (1) and (2) of Theorem 8 are satisfied if we choose , where describes the distance of each point in to the boundary . Hence, Theorem 8 gives the desired eventual positivity. ∎
Remark 10. (a) The argument sketched above breaks down for general domains in , since the inverse need no longer be positive in this case. This is a very well-studied topic in PDE theory; see for instance the surveys  by Sweers and  by Dall’Acqua and Sweers for more information.
(b) However, if we replace with a domain which is sufficiently close to a ball, we still obtain the same result. The main point here is that positivity of or, under slightly larger perturbations, at least positivity of the leading eigenfunction of remains true on such domains as shown by Grunau and Sweers in [27, Theorem 5.2]. So Theorem 9 holds on this more general class of domains, too.
(d) If we replace the clamped plate boundary conditions with so-called hinged boundary conditions, which require on the boundary, the situation becomes much easier because the operator can then be written as minus the square of the Dirichlet Laplace operator. In this case, we have eventual positivity of the semigroup on general domains; on the space of continuous functions, this example is worked out in [16, Theorem 6.1].
Non-local boundary conditions
Let us now go back to the unit interval and consider the Laplace operator, i.e. the second spatial derivative. If we impose local boundary conditions – such as for instance Dirichlet, Neumann or mixed Dirichlet and Neumann boundary condition, the Laplace operator is well-known to generate a positive semigroup (also on general domains in arbitrary dimension); see for instance [30, Corollary 4.3]. However, let us consider an example of non-local boundary conditions instead. More specifically, we consider the operator on given by
This is a self-adjoint operator; the operator, and in particular its relation to the Dirichlet and the Neumann Laplace operator, is discussed in more detail in [2, Section 3]. The spectral bound of is negative, and the inverse can be computed explicitly [16, proof of Theorem 6.11 (i)]; from this formula and the spectral theory of positive operators, we can conclude that is a simple eigenvalue and that there is a corresponding eigenfunction which is strictly positive on the closed interval ; see [16, Theorem 6.11] for details. Moreover, we have , so the assumptions of Theorem 8 are satisfied for , and we obtain the following result.
The semigroup on generated by the Laplace operator with the non-local boundary conditions given above is uniformly eventually positive.
Compare also [4, Section 4.2] for a related discussion. An example of eventual positivity for different non-local boundary conditions which lead to a non-self-adjoint realisation of the Laplace operator can be found in [14, Theorem 4.3].
Today, eventual positivity, and closely related properties as for instance asymptotic positivity, are known for various further -semigroups, including the semigroup generated by the Dirichlet-to-Neumann operator on the unit circle for various parameter choices  (which was the initial motivation for the development of the general theory), several delay differential equations ([17, Section 6.5], [24, Section 11.6] and [14, Theorem 4.6]), the semigroup generated by a bi-Laplacian with certain Wentzell boundary conditions [19, Section 7], various semigroups on metric graphs ([26, Proposition 3.7], [25, Section 6] and [8, Proposition 5.5]) and semigroups generated by Laplacians coupled by point interactions [28, Proposition 2].
5 Unbounded domains and local properties
The biharmonic heat equations on unbounded domains
A major drawback of Theorem 8 is that it can only be applied if the leading spectral value is even an eigenvalue of the operator . This makes it impossible to apply the theorem to various differential operators that live on unbounded domains. For instance, consider the biharmonic operator on given by
The spectrum of , which is the set , does not contain eigenvalues, so Theorem 8 cannot be applied. Still, the semigroup exhibits a certain local eventual positivity property: for every compact set and every initial value , the solution to the biharmonic heat equation becomes eventually positive on . This was proved, under slightly different assumptions on in [23, Theorem 1 (i)] and [22, Theorem 1.1 (ii)] by explicit kernel estimates; for more general powers of , a similar result was recently shown in [21, Theorem 1.1]. Under the assumptions described above, the result was proved by Fourier transform methods in [18, Theorem 2.1].
If we replace the whole space with an infinite cylinder – for instance of the form , where is a ball – and again impose clamped plate boundary conditions, the same local eventual positivity result remains true. The proof is technically more involved, though, and relies on a detailed analysis of the specific partial differential equation under consideration; see [18, Theorem 2.3 and Section 4].
However, despite the successful analysis of the aforementioned concrete differential equations, an abstract and general theory as outlined in Section 3 for operators with leading eigenvalue is not yet in sight for the case without eigenvalues.
Open problem. Develop a theory of locally eventually positive -semigroups which is applicable in situations where the generator does not have a leading eigenvalue.
Getting back to operators which do have a leading eigenvalue, results such as Theorem 8 might still not be applicable in some cases due to conditions (1) and (2) which are sometimes particularly subtle at the boundary of (if is, say, a domain in and a differential operator). When all functions are restricted to compact subsets of , though, conditions of the type (1) and (2) might still be satisfied.
This motivates the development of a theory of locally eventually positive semigroups for generators that do have a leading eigenvalue with strictly positive eigenfunction. Such a theory was presented by Arora in . An application of the theory to certain fourth order operators with unbounded coefficients on (which sometimes have eigenvalues due to the growth of the coefficients) was given in [1, Section 3.2].
6 Related topics and results
We close the article by discussing a few related concepts.
If generates a positive -semigroup on a function space , it is quite easy to see that if is a positive and bounded linear operator on and is a bounded and real-valued multiplication operator on , then the perturbed semigroup is positive, too: if , this follows for instance from the so-called Dyson–Phillips series representation of perturbed semigroups [20, Theorem III.1.10], and if is non-zero, it follows from the previous case by using the formula
for a real number that is sufficiently large to ensure that is positive.
For eventual positivity, though, the situation is much more subtle. Under quite general conditions, one can show that eventual positivity of a semigroup cannot be preserved by all positive perturbations of the generator. This was proved in [15, Theorem 2.3]; related results in finite dimensions had earlier been obtained in [32, Theorem 3.5 and Proposition 3.6]. On the other hand, sufficiently small positive perturbations can be shown not to destroy eventual positivity under appropriate assumptions [15, Section 4].
Maximum and anti-maximum principles
One abstract way to formulate that a linear operator on a function space satisfies a maximum principle is to require that be a positive operator, i.e. maps positive functions to positive functions. If is in the spectrum of , or more generally if the spectral bound of satisfies , it is often more natural to consider the resolvent for real numbers . If the resolvent at one such point is positive, then the same is true for all , too, and we say that satisfies a maximum principle. More precisely, this is a uniform maximum principle, while we say that satisfies an individual maximum principle if, for each , there exists an (-dependent) number such that for all .
Similarly, it is common to say that satisfies a uniform anti-maximum principle if is, say, an isolated spectral value and for all in a left neighbourhood of the resolvent maps positive functions to negative functions. Likewise, we can define an individual anti-maximum principle (and clearly, the same concepts can be defined at isolated spectral values different from , too).
Anti-maximum principles have a considerable history and have, for instance, been studied for various elliptic differential operators; see e.g.  for a seminal paper on this topic. For biharmonic and polyharmonic operators the validity of (anti-)maximum principles is closely related to the boundary conditions and the geometry of the underlying domain, as explained in Remark 10.
The argument sketched after Theorem 9 can be generalised (and partially reversed) to obtain a correspondence between the following three types of properties:
eventual positivity of the semigroup ,
spectral properties of and positivity of the leading eigenfunction,
an individual (anti-)maximum principle for .
This correspondence was discussed in [16, Sections 3–5], where the terminology eventual positivity and negativity of the resolvent was used to describe maximum and anti-maximum principles. Indeed, equivalence between the three properties (a)–(c) is true under a number of technical restrictions which have been analysed in more detail in .
Uniform (anti-)maximum principles are more difficult to analyse than their individual counterparts – a phenomenon that occurs, as pointed out above, for semigroups, too, but becomes even more pronounced when studying (anti-)maximum principles. An abstract operator theoretic approach to uniform anti-maximum principles was first presented by Takáč in [34, Section 5], and recent progress on the topic was made in . As a sample result, let us discuss the following special case of [7, Corollary 5.4] for self-adjoint operators on .
Let be a -finite measure space, and let be a real and self-adjoint operator on . Let be a function which is almost everywhere, and assume that there exists an integer such that every vector in is dominated in modulus by a multiple of . Assume moreover that is an isolated spectral value of and a simple eigenvalue whose eigenspace contains a function that satisfies for a number .
If is in the resolvent set of and , then the following assertions are equivalent.
One has for all in a left neighbourhood of .
There exists a real number such that
The assumption that be a real operator means that the domain is spanned by real-valued functions and that maps real-valued functions to real-valued functions. Assertion (i) of the theorem is a uniform anti-maximum principle, while assertion (ii) can be considered as an upper kernel estimate for the resolvent (in other words: as an upper Green function estimate) of . Simple consequences of this theorem are the classical results that the Dirichlet Laplace operator on an interval does not satisfy a uniform anti-maximum principle, while the Neumann Laplace operator on an interval does (see [7, Proposition 6.1 (a) and (b)] for a few more details). More involved examples where the theorem (or more general versions thereof) can be applied are discussed in [7, Section 6].
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Cite this article
Jochen Glück, Evolution equations with eventually positive solutions. Eur. Math. Soc. Mag. 123 (2022), pp. 4–11DOI 10.4171/MAG/65