Convergence to equilibrium for linear parabolic systems coupled by matrix-valued potentials

We consider systems of parabolic linear equations, subject to Neumann boundary conditions on bounded domains in ℝ 𝑑 , that are coupled by a matrix-valued potential 𝑉 , and investigate under which conditions each solution to such a sys-tem converges to an equilibrium as 𝑡 → ∞ . While this is clearly a fundamental question about systems of parabolic equations, it has been studied, up to now, only under certain positivity assumptions on the potential 𝑉 . Without positivity, Perron–Frobenius theory cannot be applied and the problem is seemingly wide open. In this paper, we address this problem for all potentials that are 𝓁 𝑝 - dissipative for some 𝑝 ∈ [1, ∞] . While the case 𝑝 = 2 can be treated by classical Hilbert space methods, the matter becomes more delicate for 𝑝 ≠ 2 . We solve this problem by employing recent spectral theoretic results that are closely tied to the geometric structure of 𝐿 𝑝 -spaces.


Coupled parabolic equations
On a bounded domain Ω ⊆ ℝ  with sufficiently smooth boundary, consider elliptic differential operators  1 , … ,   in divergence form with Neumann boundary conditions and a bounded measurable function  ∶ Ω → ℂ × .In this paper, we are interested in the long-time behavior of the solutions to the coupled parabolic equation .
(1.1)Such equations arise, for instance, as linearizations of reaction-diffusion equations; this renders uniform convergence of the solutions to 0, a property of (1.1) that is particularly desirable, since this property is related (by the principle of linearized stability, see, for instance, [8,Theorem 10.2.2]) to asymptotic stability of equilibria of reaction-diffusion equations.Asymptotic stability of equilibria has been studied for many specific classes of reaction-diffusion equations and by various methods; see, for instance, [18, section 3], [27,Theorem 3], [26,Theorem 1.1] (and, e.g., [2,Theorem 1.4] for an equation with Robin rather than Neumann boundary conditions).
On the other hand, there appears to be a remarkable gap in the literature concerning the convergence of the solutions to the linear equation (1.1), in its general form, to possibly nonzero equilibria.The authors are aware of only three convergence results for equations similar to (1.1): (1) In [24,Sections 7 and 8], the long-term behavior of such systems (though in nondivergence form) was considered with Dirichlet boundary conditions and under assumptions on the potential that ensure positivity of the solution semigroup.The arguments to obtain convergence to equilibrium rely on Perron-Frobenius theory.(2) In [1,Theorem 4.4], parabolic systems on the whole space ℝ  are studied, also under certain positivity assumption on the potential and by employing arguments from Perron-Frobenius theory.(3) In [10,Section 5], the present authors used recent spectral theoretic results from [16] to discuss the long-term behavior of solutions to parabolic systems on the whole space ℝ  for  ∞ -dissipative potentials.

Difficulties in the analysis of the long-term behavior
Heuristically, the long-term behavior of (1.1) is more involved than in the scalar-valued case since, even in cases where the potential  has only real entries and is assumed to preserve boundedness of all solutions, the coupling by  can cause the existence of periodic solutions.This is, for instance, the case in the simple example where  = 2,  1 ∶=  2 ∶= Δ and () ∶= ) for all  ∈ Ω.
In this example, dissipativity of  with respect to the Euclidean norm on ℝ 2 implies that all solutions to (1.1) are bounded (see Section 2 for details).However, is a periodic solution to (1.1) (where  denotes the constant function on Ω with value 1).This kind of behavior is not difficult to understand for potentials , which do not depend on the spatial variable and for elliptic operators  1 , … ,   , which all coincide (see Section 3.4 for details).However, the matter becomes more delicate if one of these two properties is not satisfied.

Contributions in this paper
The purpose of this paper is to give a variety of conditions for the convergence to equilibrium of the solutions to (1.1).After some preliminaries in Section 2, several convergence criteria are established in Section 3. The third of these criteria is probably the most surprising one, and it is particularly interesting for the cases  = 1 and  = ∞ in which the condition is easily checkable.(e) If all elliptic operators   coincide and the matrices () are simultaneously diagonalizable, then the equation can be decomposed into scalar-valued subsystems, for which sufficient conditions for convergence are easier to find.
The main results are criteria (a)-(c), which will be presented in the Sections 3.1-3.3,respectively.The remaining criteria (d) and (e) presented in Sections 3.4 and 3.5 are fairly specific.They are mainly included to cover particularly simple cases where the long-term behavior can be analyzed by easier and more direct methods than in cases (a)-(c).
Let us give a brief overview of the different tools, which we will use to prove each of the five criteria (a)-(e): (a) This criterion will be derived from Perron-Frobenius theory for positive  0 -semigroups.
(b) This follows from elementary dissipativity estimates.
(c) This criterion will be derived from results in [10,16], which relate the spectrum of contractive semigroups to the geometry of   -spaces.(d) This is mainly an exercise in linear algebra.(e) This criterion follows from combining a diagonalization argument with a special case of (b).
We stress that no new theoretical methods are developed in this paper.The main point of this paper is rather to combine various abstract theorems from the literature to understand the long-term behavior of (1.1) in a large variety of cases.

Further related literature
As explained before, equations of the type (1.1) are closely related to reaction-diffusion equations.In addition, there is a branch of the literature where heat equations coupled by matrix-valued potentials are studied with the following focus: For instance, in [19,20,23], equations on the whole space ℝ  are considered where the potential  is not assumed to be bounded.In this case, a major point of interest is well-posedness of the equation, that is, whether the differential operator generates a  0 -semigroup.In contrast, our setting is simpler in the sense that generation properties follow from elementary perturbation theory; instead, our main interest is to give nontrivial conditions on the potential that imply convergence to equilibrium.

Organization of the paper
In Section 2, we discuss the detailed setting in which the parabolic problem (1.1) is posed.Further, it is explained how the solution semigroup acts on the   -scale.The results from this section are needed as a proper set-up for our convergence results in Section 3. In the Appendix, we recall characterizations of -dissipativity for finite-dimensional matrices.

The equation
Let ∅ ≠ Ω ⊆ ℝ  be a bounded domain, which has the extension property in the sense that every Sobolev function in  1 (Ω; ℂ) is the restriction of a Sobolev function in  1 (ℝ  ; ℂ).This is the case, for example, if Ω has Lipschitz boundary [3,Section 7.3.6].
We fix an integer  ≥ 1 (which will denote the number of coupled equations on Ω) as well as measurable and bounded functions  1 , … ,   ∶ Ω → ℝ × and  ∶ Ω → ℂ × .Moreover, we assume that there exists a constant  > 0 such that for all  ∈ {1, … , } and almost all  ∈ Ω, the uniform coercivity condition holds for all  ∈ ℂ  .We will study the long-term behavior of the solutions to the coupled parabolic equation that is formally given by subject to Neumann boundary conditions.Due to the weak regularity assumptions on the coefficients and on the boundary of Ω, we use form methods to give precise meaning to the elliptic operators  ↦ div(  ∇): For each  ∈ {1, … , }, we define a bilinear form This form induces a linear operator −  ∶  2 (Ω; ℂ) ⊇ (  ) →  2 (Ω; ℂ), and   is interpreted as a realization of the differential operator  ↦ div(  ∇) with Neumann boundary conditions.Moreover, each operator   generates a positive (in the sense of Banach lattices) and contractive  0 -semigroup (e   ) ≥0 on  2 (Ω; ℂ).For a general overview of form methods in the context of heat equations, we refer the reader to [25].

Behavior on the 𝑳 𝒑 -scale
In this subsection, we briefly discuss how those semigroups act on the   -scale.We will see that, due to an ultracontractivity argument, most of the relevant properties do not depend on the choice of .The arguments in this subsection are fairly standard, but there are a few subtleties since we also want to consider the respective semigroups for  = ∞.Thus, all the relevant properties of the semigroups are stated in detail.
We start with the observation that the constant function  is a fixed point of each semigroup (e   ) ≥0 and its dual.Thus, it follows from interpolation theory that these semigroups induce positive and contractive  0 -semigroups on the   -scale, where  ∈ [1, ∞).We denote the generators corresponding to those semigroups by  , .In particular, one has  ,2 =   .
3) is of course equivalent to the norm that we would obtain by endowing ℂ  with the Euclidean norm and then endowing   (Ω; ℂ  ) with the vector-valued -norm.However, the main advantage of the norm ‖ ⋅ ‖  defined in (2.3) is that it renders   (Ω; ℂ  ) isometrically lattice isomorphic to the   -space of scalar-valued functions over  disjoint copies of Ω, that is, we can treat   (Ω; ℂ  ) as a scalar-valued   -space.In particular, the geometry of   (Ω; ℂ  ) coincides with that of a scalar-valued   -space.
In what follows, we will use the symbol  both to denote the function  ∶ Ω → ℂ × that was introduced in the previous subsection and the operator   (Ω; ℂ  ) →   (Ω; ℂ  ) given by multiplication with this function (for any  ∈ [1, ∞]).
The semigroups (e (  +) ) ≥0 are consistent on the   -scale.This follows from a perturbation argument (e.g., by utilizing Trotter's product formula or the Dyson-Phillips series) since the semigroups generated by   are consistent.
Moreover,  ∞ (Ω; ℂ  ) is invariant under the action of the semigroups (e (  +) ) ≥0 as the following proposition shows.For a proper reading of the proposition, note that the realizations of the multiplication operator  as bounded operators on   (Ω; ℂ  ) are consistent for  ∈ [1, ∞].Moreover, the exponential operators e  are, for every  ≥ 0, also consistent on the   (Ω; ℂ  )-scale; in other words, for 1 ≤  ≤  ≤ ∞, it does not make a difference whether we consider the exponential e  on   (Ω; ℂ  ) first and then restrict it to   (Ω; ℂ  ) or whether we consider it on   (Ω; ℂ  ) in the first place.
Proof.The existence of  follows from the fact that ‖ ‖ e  ‖ ‖ ∞→∞ ≤ e ‖‖ ∞→∞ for all  ≥ 0. Now, fix such an  as well as  ≥ 0 and  ∈ [1, ∞).By Trotter's product formula (cf.[12, Corollary III.V.8]), we have e (  +)  = lim →∞ (e   e    )   with respect to the   -norm for each  ∈   .The semigroup generated by   e    )   is an element of e  B[0, 1] for each  ∈ ℕ and so is the limit as Since the semigroups act consistently on the   -scale, the restriction of the operator e (  +) to  ∞ (Ω; ℂ  ) is the same operator for all  ∈ [1, ∞).From now on, by abuse of notation, the restriction of e (  +) to  ∞ (Ω; ℂ  ) is denoted by e ( ∞ +) .Note that this is used purely as a notation.In particular, no operator  ∞ is defined and no assertions about such an operator are made.Clearly, (e ( ∞ +) ) ≥0 is an operator semigroup, but it is certainly not strongly continuous in general.However, it follows from Proposition 2.3 below that this semigroup is strongly continuous and, in fact, even norm continuous on the open time interval (0, ∞).
For  = ∞, observe that e ( ∞ +) factors as Hence, the claim follows from the case  = 2. □ As two consequences of the above proposition, boundedness and operator norm convergence of the semigroup do not depend on the choice of .
For  ≥ 2, the operator e (  +) ∶=   →  ∞ factors as Therefore, e (  +) converges in (  ;  ∞ ) with respect to the operator norm as  → ∞. □ Corollaries 2.4 and 2.5 show that if the solutions to the coupled Cauchy problem (2.2) are bounded or converge uniformly in one -norm, they are bounded or converge uniformly in every -norm, respectively.This motivates the following terminology that will be used throughout the rest of the paper.Definition 2.6.
(a) We say that the solutions to the coupled heat equation (2.2) are uniformly bounded if one, and thus all, of the equivalent assertions from Corollary 2.4 are satisfied.(b) We say that the solutions to the coupled heat equation (2.2) converge uniformly as  → ∞ if one, and thus all, of the equivalent assertions of Corollary 2.5 are satisfied.
In Section 3, we provide sufficient conditions for the uniform convergence in the sense of Definition 2.6 of the solutions to (2.2) as  → ∞.

Dissipativity
In view of Corollary 2.4, boundedness of the solution semigroup to (2.2) for one  implies boundedness on the entire   -scale.The easiest way to obtain boundedness for some  ∈ [1, ∞] is to assume that the multiplication operator  is dissipative on   (Ω; ℂ  ).This is discussed in detail in this subsection.For a general treatment of the concept dissipativity, including its definition and various characterizations, we refer, for instance, to [12, Section II.If, in addition, () ∈ ℝ × almost all  ∈ Ω, then the above assertions are also equivalent to the following: (iii) For almost all  ∈ Ω, the matrix () is dissipative with respect to the   -norm on ℝ  .
Proof.The equivalence of (i) and (ii) is an immediate consequence of our choice of the norm on   (Ω; ℂ  ) (see formula (2.3)).Now assume that the matrix () has only real entries for almost all  ∈ Ω.The implication from (ii) to (iii) is obvious.To show that (iii) implies (ii) note that, for every matrix  ∈ ℝ × , its operator norm induced by the -norm on ℝ  coincides with its operator norm induced by the -norm on ℂ  .Indeed, for  ∈ [1, ∞), this is [14, Proposition 2.1.1],and for  = ∞, this follows from the identity So overall, if e () is contractive on (ℝ  , ‖ ⋅ ‖  ), then it is also contractive on (ℂ  , ‖ ⋅ ‖  ), that is, (iii) implies (ii).□ In view of Proposition 2.8, it is worthwhile to recall that dissipativity of matrices with respect to the   -norm on ℝ  can be characterized quite explicitly.For the convenience of the reader, the said characterization is presented in Proposition A.1 in the Appendix.

CONVERGENCE TO EQUILIBRIUM
In this section, we prove several criteria under which the solutions to the coupled heat equation (2.2) converge uniformly in the sense of Definition 2.6 (b).In order to provide a simple but illuminating illustration and comparison of our results, in each subsection, the long-term behavior of the solutions to a concrete, simple toy example is discussed.Namely, we consider the evolution equation for a potential  ∶ Ω → ℝ 2×2 subject to Neumann boundary conditions on Ω.However, the results from these sections hold, of course, for much more general settings.
The following table provides an overview of the following subsections and the respective convergence results proven in each of them.First, we turn our attention to situations in which the long-term behavior of the system can be investigated by employing arguments from classical Perron-Frobenius theory.

Convergence for quasi-positive potentials
A matrix  ∈ ℝ × is called quasi-positive if all off-diagonal entries of  are ≥ 0. In this section, it is shown that the solutions to the coupled parabolic equation (2.2) converge uniformly if they are bounded and if the potential  is quasipositive almost everywhere.The key insight is that the quasi-positivity of  yields that the solution semigroup is positive.
Proof.According to Proposition 2.3 (ii), the semigroup (e ( 2 +) ) ≥0 is immediately compact.Hence, the positivity of (e ( 2 +) ) ≥0 implies that the so-called boundary spectrum If s( 2 + ) < 0, then the immediate compactness of the semigroup implies that it converges uniformly to 0 (as immediately compact semigroups satisfy the spectral mapping theorem, see, e.g., [12,Corollary IV.3.12]).So assume now that s( 2 + ) = 0.Then, according to what we just showed, 0 is the only spectral value of  2 +  on the imaginary axis; it is a pole of the resolvent (due to the immediate compactness of the semigroup) and this pole is of order 1 since the semigroup is bounded.This implies the operator norm convergence as  → ∞, see, e.g., [13,Proposition V.4.3].So, the solutions to (2.2) converge uniformly as  → ∞. □ A related result for coupled parabolic equations on the whole space ℝ  was proved in [1, Theorem 4.4].There, the matrices in the potential are supposed to be quasi-positive, but the assumptions on the differential operator and its coefficients are distinct from those in this paper.
Next, a sufficient condition for uniform boundedness of the solutions to Equation (2.2) is presented.This condition guarantees that Theorem 3.2 can be applied.Proposition 3.3.Suppose that () is in ℝ × and is quasi-positive for almost every  ∈ Ω and that there exists a vector  ∈ ℝ  whose components are all strictly greater than 0 and which satisfies () = 0 for almost all  ∈ Ω.Then, the solutions to Equation (2.2) are uniformly bounded and thus converge uniformly as  → ∞.
Moreover, each vector in the real space  ∞ (Ω; ℝ  ) is bounded above and below by a multiple of  ⊗  and thus, by the positivity of the semigroup (e ( 2 +) ) ≥0 , the orbit of the vector is bounded in  ∞ (Ω; ℝ  ).Consequently, the same is true for every vector in  ∞ (Ω; ℂ  ).Hence, the semigroup (e ( ∞ +) ) ≥0 on  ∞ (Ω; ℂ  ) is bounded by the uniform boundedness principle and Theorem 3.2 yields the claim.□ Recall that a positive  0 -semigroup on an   -space is called irreducible if the only invariant closed ideals are {0} and the entire space (see, for instance, [5, Section C-III-3] or [6, Section 14.3] for details).Irreducibility for a positive linear operator is defined analogously.If the matrix  is irreducible, then the limit projection  on   (Ω; ℂ  ) of the semigroup (e (  +) ) ≥0 is either 0 or has rank 1.
Proof.Let us first consider the case  ∈ [1, ∞) (since we have a  0 -semigroup in this case).Since Ω is connected, the heat semigroup generated by   on   (Ω; ℂ) is irreducible.As the matrix  is also irreducible, we can hence apply the perturbation result in [5, Proposition C-III-3.3] to see that the semigroup generated by  2 +  on   (Ω; ℂ  ) is also irreducible; this argument is taken from [24, Proposition in Section 8].The irreducibility of the semigroup implies that the limit operator is either 0 or has rank 1; this follows from classical arguments in Perron-Frobenius theory, see, for instance, [4, Proposition 3.1(c)] for a detailed explanation.
Since the semigroups generated by   +  act consistently on the   -scale, the same is true for the limit operator.Hence, the conclusion carries over to the case  = ∞.□ Due to the irreducibility in the preceding remark, even a bit more can actually be said about the limit operator .We refrain from discussing this in detail here and refer to the general result explained in [4, Proposition 3.1(c)] instead.We conclude this subsection with a simple example.
Then each matrix () is quasi-positive and the vector  ∶= (2, 1) T is in the kernel of each ().Thus, it follows from Proposition 3.3 that the solutions to the evolution equation (3.1) converge uniformly as  → ∞.Since the function  0 ∶=  ⊗  is an equilibrium, the limit is nonzero for some initial values.Remark 3.4 shows that if at least one of the functions  and  is nonzero on a set of strictly positive measures, then the limit operator has rank 1.
One interesting aspect of the potential matrices () in this example is that they are, in general, not simultaneously diagonalizable (and hence, the results from Section 3.5 below cannot be applied); this follows from the fact that the respective second eigenspaces (the ones not spanned by ) of the matrices

Convergence for 𝓵 𝟐 -dissipative potentials
In this subsection, convergence of the solutions to the coupled parabolic equation (2.2) is characterized for the case where the matrices () are  2 -dissipative.
Proposition 3.6.Suppose that, for almost all  ∈ Ω, the matrix () is dissipative with respect to the  2 -norm on ℂ  .For each i ∈ iℝ, the following two assertions are equivalent: (i) The number i is in the point spectrum  pnt ( 2 + ).
(ii) There exists a measurable subset Ω ⊆ Ω of full measure (i.e., the difference Ω ⧵ Ω has Lebesgue measure 0) such that In this case, each component function of every eigenvector  ∈ ker (i − ( 2 + )) is constant almost everywhere on Ω. Proof.

𝑉(𝑥)𝑧 = 𝑉(𝑥)𝑢(𝑥) = i𝛽𝑢(𝑥) = i𝛽𝑧
for almost every  ∈ Ω.Consequently, there exists a measurable set Ω ⊆ Ω of full measure such that  ∈ ⋂ ∈ Ω ker(i − ()).□ The previous proposition characterizes, in terms of the matrices (), whether  2 +  has a nonzero imaginary eigenvalue.This yields the following characterization of uniform convergence for the solutions to the coupled heat equation (2.2).Theorem 3.7.Assume that, for almost all  ∈ Ω, the matrix () is dissipative with respect to the  2 -norm on ℂ  .Then, the following assertions are equivalent: (i) The solutions to the coupled heat equation (2.2) converge uniformly as  → ∞. (ii) For every i ∈ iℝ ⧵ {0} and every measurable subset Ω ⊆ Ω of full measure, we have Proof.According to Proposition 3.6, assertion (ii) of the theorem is equivalent to the assertion that  2 +  does not have any nonzero eigenvalues on the imaginary axis.So, it is left to show that this is equivalent to uniform convergence of the solutions to (2.2): (i) ⇒ (ii): If (ii) does not hold, then  2 +  has an eigenvalue i ∈ iℝ ⧵ {0} with eigenvector .Thus, e ( 2 +)  does not converge as  → ∞, which means that (i) does not hold.(ii) ⇒ (i): Conversely, suppose  2 +  does not have any eigenvalues on the imaginary axis, except for possibly 0.
According to Proposition 2.3 (ii), the semigroup (e ( 2 +) ) ≥0 is immediately compact.Moreover, it is contractive as  2 +  is dissipative.Thus, the same spectral theoretic argument is in the proof of Theorem 3.2 and implies that the solutions to (2.2) converge uniformly as  → ∞. □ Let us state the following special case of Theorem 3.7 explicitly.
Corollary 3.8.Suppose that, for almost all  ∈ Ω, the matrix () is dissipative with respect to the  2 -norm on ℂ  .If
Example 3.9.Let  = 2, let  ∶ Ω → ℝ ⧵ {0} be bounded and measurable, and let the potential  be given by This potential is not quasi-positive but, since each matrix () is  2 -dissipative, all solutions to the evolution equation (3.1) are bounded on   (Ω) for any  ∈ [1, ∞] (see Corollary 2.4).The spectrum of each matrix () is {−i(), i()}, so it follows from Theorem 3.7 that all solutions to (3.1) convergence uniformly as  → ∞ if and only if  is not constant almost everywhere.

Convergence for 𝓵 𝒑 -dissipative potentials
A drawback of the techniques employed in the preceding section is that they rely heavily on the 2-dissipativity of the matrices ().In this section, we will instead assume that the matrices () are dissipative with respect to the   -norm for some fixed  ∈ [1, ∞],  ≠ 2, and have real entries only.As Proposition 3.10 below shows, this assumption is stronger than assuming (()) ∩ iℝ ⊆ {0} for almost all  ∈ Ω.As can be seen in the simple Example 3.13, there are cases where -dissipativity of the () is satisfied for some  ≠ 2 while 2-dissipativity is not.
For our analysis, we need spectral theoretic results on a class of spaces that we call projectively non-Hilbert spaces.This notion is taken from [16, Definition 3.1]: A real Banach space  is called projectively non-Hilbert if for no contractive rank-2 projection  ∈ (), the range  is isometrically isomorph to a Hilbert space.This is, for instance, the case for each realvalued   -space,  ∈ [1, ∞] ⧵ {2} (cf.[16, Example 3.2] and the discussion after [16,Example 3.5]; in the finite-dimensional case, this was already observed in [21, Propositions 1 and 2]).
The following proposition is a finite-dimensional special case of [16,Theorem 3.7].
We point out that the main idea that underlies this proposition is much older and goes back to Lyubich [21, Theorem 1] (see also [7,Section 2.4] and [22,Corollary 3.9]) who formulated a closely related result in the discrete-time case.
Note that the assertion of Proposition 3.10 fails in the case  = 2.For example, consider the matrix ) This matrix is dissipative with respect to the  2 -norm on ℝ 2 , but () = {−i, i}.Moreover, we stress that it is essential in Proposition 3.10 that the matrices () have only real entries (otherwise, the operator i on the one-dimensional space ℂ is a counterexample).
For  = ∞, we do not have a  0 -semigroup.However, we point out that we consider the case  = ∞ to be quite significant (rather than just an interesting side note) since the assumption that () be dissipative with respect to the   -norm is easiest to check if  is either 1 or ∞, see Proposition A.1.
The following simple example shows that there are situations where Theorem 3.11 can be applied, while the other results of Section 3 cannot.Example 3.13.Let  = 2, let ,  ∶ Ω → [0, ∞) measurable and bounded, and let the potential  be given by for all  ∈ Ω.
Each matrix () is  ∞ -dissipative (Proposition A.1).So it follows from Theorem 3.11 that the solutions to the evolution equation (3.1) converge uniformly as  → ∞.The function (, − ) T is an equilibrium, so the limit is nonzero for some initial values.Again, we note that the matrices () are not simultaneously diagonalizable in general, since the matrices have different sets of eigenvectors.So the system cannot be uncoupled by diagonalization, in general.Moreover, we note that the matrices () are not  2 -dissipative in general, since a short computation shows that the symmetric part of the matrix ( − − − − ) always has a strictly positive eigenvalue if ,  ∈ (0, ∞) are two distinct numbers.Hence, the Hilbert space technique from Section 3.2 is not applicable.

Constant potentials
In this section, we discuss a setting in which the special algebraic structure of the coupled equation allows us to make use of an ad hoc argument in order to determine the long-term behavior.In particular, one can show in this case that the long-term behavior of the system is governed solely by the spectral properties of the potential.
In this section, we study the situation where (a)  1 () = ⋯ =   () for almost all  ∈ Ω, (b)  is constant almost everywhere.
As  is constant almost everywhere, there is a unique matrix, which coincides with the function ; and abusing the notation, we denote this matrix again by .This means that there are two semigroups related to : (1) On one hand, (e  ) ≥0 defines a semigroup on ℂ  .
It is easy to see that ‖ ‖ e  ‖ ‖ 2,ℂ  →ℂ  = ‖ ‖ e  ‖ ‖  2 → 2 for all  ≥ 0 (where the former norm is the one induced by the Euclidean norm on ℂ  ).
The next lemma shows that for constant potentials , the semigroup (e ( 2 +) ) ≥0 is given by a tensor product of the semigroup generated by  on ℂ  and the semigroup generated by  1 = ⋯ =   on  2 (Ω; ℂ).Lemma 3.14.If the potential  is constant almost everywhere and  1 () = ⋯ =   () for almost all  ∈ Ω, then one has e ( 2 +) = e  2 e  = e  e  2  for all  ≥ 0.
Proof.As  is constant almost everywhere and the operators on the diagonal of  2 all coincide, it is easy to see that the operators  2 and  commute on  2 (Ω; ℂ  ), which is in turn equivalent to the statement that the resolvents of both operator commute.So as a consequence of the Post-Widder inversion formula (cf.[12, Corollary III.5.5]), this implies that the semigroups (e  2 ) ≥0 and (e  ) ≥0 commute on  2 (Ω; ℂ  ).Therefore, it follows from Trotter's product formula that e ( 2 +) = e  2 e  = e  e  2 for all  ≥ 0. □ As a consequence of Lemma 3.14, the long-term behavior of the semigroup (e ( 2 +) ) ≥0 on  2 (Ω) depends solely on the asymptotic behavior of the semigroup (e  ) ≥0 on ℂ  .(i) ⇒ (ii): Since each of the (identical) semigroups (e   ) ≥0 converges with respect to the operator norm as  → ∞ (on   (Ω; ℂ) for any  ∈ [1, ∞]), so does the semigroup e  2 (on   (Ω; ℂ  ) for any  ∈ [1, ∞]).Hence, this implication is an immediate consequence of Lemma 3.14.(ii) ⇒ (i): Suppose that e  does not converge on ℂ  as  → ∞.Then, there exists some vector  ∈ ℂ  such that e   does not converge as  → ∞.Now consider the function  ⊗  ∶= ( 1 , … ,   ) T ∶ Ω → ℂ  .By Lemma 3.14, one has e ( 2 +) ( ⊗ ) = e  e  2 ( ⊗ ) = e  ( ⊗ ) =  ⊗ (e  ), for all  ≥ 0, which does not converge as  → ∞. □ As a simple example, we consider a similar potential as in Example 3.9 -and now we assume that the potential is constant, but we also allow for the case that it is 0. ) for all  ∈ Ω.
As in Example 3.9, the  2 -dissipativity of the matrix  implies that all solutions to the evolution equation ( Of course, for this specific example, the same conclusion can also be derived from Theorem 3.7.
As a result, the operators  1 , … ,   coincide and, for the sake of notational simplicity, those operators will all be denoted by .Assumptions (a) and (b) allow us to decouple the system (2.2) since  2  −1 =  2 .This means that (2. for all  = 1, … , -and by applying Theorem 3.7 to the scalar-valued case  = 1 (which is possible due to assumption (c) on the eigenvalues   ()), we see that for any  ∈ {1, … , }, the solutions to the latter equation converge if and only if there is no i ∈ iℝ ⧵ {0}, which is equal to   almost everywhere.Thus, one has the following result.
Proposition 3.17.Let the assumptions (a)-(c) from the beginning of Section 3.5 be satisfied.Then the following assertions are equivalent: (i) The solutions to the coupled heat equation (2.2) converge uniformly as  → ∞.
(ii) For each  ∈ {1, … , }, the following holds: There does not exist a number i ∈ iℝ ⧵ {0}, which is equal to the function   almost everywhere.
We note that Example 3.9 can also be treated by utilizing Proposition 3.17 since the matrices () in the example are simultaneously diagonalizable (and since the operators  1 and  2 in Equation (3.1) are both equal to the Laplace operator and thus coincide).Another simple example is the following: Example 3.18.Let  = 2, let  ∶ Ω → ℂ be a bounded and measurable function that satisfies Re () ≥ 0 for almost all  ∈ Ω.Let the potential  be given by () ∶= −() ) for all  ∈ Ω.
Then, all the matrices () are simultaneously diagonalizable and their eigenvalue curves are given by  1 () = 0 and  2 () = −3() for all  ∈ Ω.So  1 is constant, but its value is not in iℝ ⧵ {0}.Hence, Proposition 3.17 shows the following: If  is almost everywhere constant and equal to an element of iℝ ⧵ {0}, then the solutions to (3.1) do not converge as  → ∞.In all other cases, the solutions to (3.1) converge uniformly as  → ∞.
Concluding remarks.Clearly, much more remains to be done in the case that the potential  is not dissipative, since most methods presented in this paper do not work in this case.In fact, it is not even clear to the authors in general how to check boundedness of the solutions to (2.2) if  is not dissipative with respect to any   -norm on ℂ  (for one exception, though, see Proposition 3.3).
Another direction of generalization is led by the idea to consider compact Riemannian manifolds in place of the bounded domain Ω ⊆ ℝ  .
Finally, in view of the coupled first-order equations considered in [11], the question arises what happens, in general, if nonelliptic differential operators are coupled by a matrix-valued potential.

A C K N O W L E D G M E N T S
We are indebted to the referee for pointing out how to prove Proposition 3.6 without the additional assumption that the diffusion coefficients   () be symmetric.We thank Abdelaziz Rhandi for pointing out the diagonalization argument outlined in Section 3.5.Furthermore, we are indebted to Fabian Wirth for bringing Ref. [7] to our attention.Moreover, Alexander Dobrick thanks Florian Pannasch for a fruitful discussion on results from [9] regarding the theory of bicontinuous semigroups.
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( a )
If all solutions to (1.1) are bounded (say, in   (Ω; ℂ  ) for some  ∈ [1, ∞]), each matrix () is real, and the offdiagonal entries of each matrix () are positive, then every solution to (1.1) converges as  → ∞ (Section 3.1).(b) If each matrix () is  2 -dissipative on ℂ  , convergence of the solutions can be characterized purely in terms of the spectral structure of the matrices () (Section 3.2).(c) If each matrix () is real and   -dissipative on ℝ  for a fixed number  ∈ [1, ∞] ⧵ {2}, all solutions converge as  → ∞ (Section 3.3).(d) If all elliptic operators   coincide and the matrices () do not depend on , the vector-valued differential operator commutes with the coupling potential.The long-term behavior of the solutions to (1.1) can thus be characterized in terms of the long-term behavior of the semigroup on ℂ  generated by .

Proposition 3 . 15 .
Suppose that the potential  is constant almost everywhere, and that  1 () = ⋯ =   () for almost all  ∈ Ω.Then the following assertions are equivalent:(i) The matrices e  converge on ℂ  as  → ∞.(ii)The solutions to the coupled heat equation (2.2) converge uniformly as  → ∞.Proof.