STABILITY IMPLIES CONSTANCY FOR FULLY AUTONOMOUS REACTION–DIFFUSION–EQUATIONS ON FINITE METRIC GRAPHS

. We show that there are no stable stationary nonconstant solutions of the evolution problem (1) for fully autonomous reaction–diﬀusion–equations on the edges of a ﬁnite metric graph G under continuity and Kirchhoﬀ ﬂow transition conditions at the vertices.


(Communicated by Enrique Zuazua)
Abstract. We show that there are no stable stationary nonconstant solutions of the evolution problem (1) for fully autonomous reaction-diffusion-equations on the edges of a finite metric graph G under continuity and Kirchhoff flow transition conditions at the vertices. (1) u ∈ C(G × [0, ∞)) ∩ C 2,1 K (G × (0, ∞)), ∂tu j = ∂ 2 j u j + f (u j ) on the edges k j , 1. Introduction. In a fundamental paper of 1979 H. Matano [18] showed that autonomous reaction-diffusion equations involving the Laplacian under Neumann boundary conditions in a convex C 3 -domain cannot possess stable (spatially) nonconstant stationary solutions. The present paper deals with the non existence of stable nonconstant stationary solutions of reaction-diffusion-equations on the edges of a metric graph and natural transition conditions at the vertices. The parabolic problem in question reads u ∈ C(G × [0, ∞)) ∩ C 2,1 K (G × (0, ∞)), where {k j 1 ≤ j ≤ N } denotes the edge set of the metric graph G and where the subscript K stands for the validity of the Kirchhoff condition at each vertex v i of the graph. Note that we do not require the coefficients in (2) to be consistent with the diffusion coefficient, i.e. we do not impose c ij = 1, but only the dissipativity assuring condition that all c ij > 0. At the ramification nodes we impose the continuity condition that clearly is contained in the condition u ∈ C(G). Throughout, the constant coefficients and nonlinearities are assumed to satisfy Now we can state the Main Theorem 1. On any finite connected metric graph all stable stationary solutions of Problem (1) are constant.
In 2001 E. Yanagida [21] published a list of five exceptional graphs that do not allow stable nonconstant stationary solutions in the consistent case of (1), see Section 5. Moreover, he established some fundamental instability tools, as the instability criterion in the presence of two different critical points in one edge. In 2015 the authors [9] showed that the assertion of the Main Theorem holds for any metric graph with sufficiently small edge lengths, as well as for any metric graph for the cubic balanced case f (u) = u − u 3 , for f (u) = η sin(u) and for some other nonlinearities. Other recent instability criteria, also for dynamical Kirchhoff conditions, can be found in [11].
At first glance, the main result might seem to be surprising in view of the existence of stable nonconstant stationary solutions on non convex domains in higher dimensions, see [13,18] and the references therein. But, as it stands, with respect to the autonomous semilinear parabolic flows defined by (1), finite metric graphs behave like one-dimensional objects, i.e. like intervals of the real line. Clearly, Problem (1) can be regarded as an abstract interaction problem on a suitable interval in the sense of Gramsch and Ali Mehmeti, see e.g. [3,7], where the node transition conditions can be read as equivalent Cauchy conditions of order 0 and 1 and identifying conditions at interior points and on the boundary of the interval. However, tame deviations from the autonomous character of the differential equations as different diffusion coefficients or edge dependent nonlinearities can lead to the existence of nonconstant stable stationary solutions, see Section 8.
The present paper is organized as follows. After some stability prerequisites and graph theoretical preliminaries in Section 2, Section 3 presents the exclusion of stable nonconstant stationary solutions on paths and circuits. The basic cutting technique that is a crucial tool in showing the main theorem, is established in Section 4 for more general reaction-diffusion-problems and states, for short, that if the metric graph G allows the stationary solution u and if it is cut at some point p = π j (x j ) with ∂ j u j (x j ) = 0, then stability of the corresponding solution on the resulting graphG is equivalent with the one of u on G. As another extension technique the edge doubling is presented at the end of this section. In Section 5 we present some elementary cases of graphs that exclude stable nonconstant stationary solutions. In Section 6 the proof of the main theorem will be given, first for trees and then for a general finite metric graph where the first case is a key tool of a recurrence procedure using the corank of the graph. In Section 7 some energy and localization results for stationary solutions in the consistent case are treated, based on the Hamiltonian system associated to the edge differential equations. Finally, Section 8 presents some examples of stable nonconstant stationary solutions of non autonomous problems as edge dependent diffusion coefficients or edge dependent nonlinearities, as well as under different node transition conditions. 2. Metric graphs and stability. For any graph Γ = (V, E, ∈), the vertex set is denoted by V = V (Γ), the edge set by E = E(Γ) and the incidence relation by ∈⊂ V × E. The valency of each vertex v is denoted by γ(v) = #{k ∈ E v ∈ k} counting incident loops twice. Unless otherwise stated, all graphs considered in this paper are assumed to be nonempty, connected and finite with n = #V, N = #E.
The vertices will be numbered by v 1 , . . . , v n , the respective valencies by γ 1 , . . . , γ n , and the edges by k 1 , . . . , k N . The boundary vertices V b = {v i ∈ V γ i = 1} will be distinguished from the ramification nodes V r = {v i ∈ V γ i ≥ 2} and the essential ramification nodes V ess = {v i ∈ V γ i ≥ 3}. By definition, a circuit is a connected and regular graph of valency 2. A path is a connected graph with two distinct vertices of valency 1 while the other vertices are all of valency 2. By definition, a viaduct π in Γ is a path in Γ joining two distinct vertices v andṽ such that there is no other walk in Γ joining v andṽ having a vertex in the set V (π)\{v,ṽ}. For further graph theoretical terminology we refer to [12,20]. Moreover, we consider each graph as a topological graph in R m , i.e. V (Γ) ⊂ R m and the edge set consists in a collection of Jordan curves with the following properties: Each support k j = π j ([0, j ]) has its endpoints in the set V (Γ), any two vertices in V (Γ) can be connected by a path with arcs in E(Γ), and any two edges k j = k h satisfy k j ∩k h ⊂ V (Γ) and #(k j ∩k h ) ≤ 2. The arc length parameter of an edge k j is denoted by x j . Unless otherwise stated, we identify the graph Γ = (V, E, ∈) with its associated metric graph, network or quantum graph especially each edge π j with its support k j . Throughout it will be assumed that all π j ∈ C 2 ([0, j ], R m ). Thus, endowed with the induced topology G is a connected and compact space in R m . Throughout, we shall denote the total graph length by The orientation of the graph Γ is given by the incidence matrix D(Γ) = (d ik ) n×N with

JOACHIM VON BELOW AND JOSÉ A. LUBARY
For a function u : G → C we set u j := u • π j : [0, j ] → C and use the abbreviations etc. and Endowed with a usual product norm we set for p ∈ [1, ∞] and k ∈ N, respectively. The validity of the Kirchhoff law (2) in a function space will be indicated by the subscript K. In particular, for m ∈ N * , r ∈ N we set , u satisfies (2) }, and for functions depending also on a time variable that ranges in an interval I, C m,r K (G × I) denotes the vector space of all the functions u ∈ C(G × I) satisfying (2) and Closing this section we recall Lyapunov's notion of stability associated to the following reaction -diffusion -problem under the hypotheses A stationary solution w of Problem (5) is called stable if for each > 0, there exists a δ > 0 such that, for each initial data u 0 ∈ C(G) with u 0 − w ∞,G < δ the solution u of (5) exists in [0, ∞) and satisfies Several instability criteria have been established in [9], among which we cite the following.
Lemma 2.1. If a stationary solution u ∈ C 2 K (G) of (5) under consistent Kirchhoff conditions satisfies either then u is unstable.
Moreover, let us cite the following elementary result from [9].
If, in addition, u is nonconstant, then the last inequality is strict.
On all finite metric graphs it can happen that there are no stationary solutions at all. E.g. for f ≡ 1, there is no such solution u ∈ C 2 K (G) of (1) with a consistent Kirchhoff law, since such a solution would have to fulfill G ∂ 2 u dx = 0 by the Kirchhoff flow condition (2).
3. Paths and circuits. In this section we establish the exclusion of stable nonconstant stationary solutions on paths and circuits. Let u ∈ C 2 K (G) be a stationary solution of the autonomous and consistent problem Recall that the Rayleigh quotient of the linearized elliptic operator at u for (7) is given by and that its admissible functions ϕ are just given by the elements of C(G) ∩ H 1 (G). Set ψ j = ∂ j u j . On each edge k j , ψ j satisfies the linearized equation by standard regularity arguments. Now let Π be a path with N edges and vertices numbered by 1, . . . , N + 1 such that v i and v i+1 are adjacent, and such that V b = {v 1 , v N +1 } forms the set of boundary vertices. The incidence matrix is given by Lemma 3.1. Suppose that u ∈ C 2 K (Π) is a nonconstant stationary solution of (7) on the path Π. Then u is unstable, more precisely Proof. Here, in fact, ψ = (ψ j ) N ×1 defines an admissible function for the Rayleigh quotient since ψ constitutes a function belonging to C 1 [0, L] by the consistent Kirchhoff condition in (7), by (3) and by the edge differential equations. Thus, denoting the numerator of R(ϕ; u) by N (ϕ), we obtain by (8) Thus, R(ψ; u) = 0 and, in turn, λ 0 (u) ≤ 0. If λ 0 (u) < 0, the instability follows from [9,21]. It remains to exclude the case λ 0 (u) = 0. But, in that case, ψ is not only admissible, but an eigenfunction belonging to the simple eigenvalue λ 0 = 0 and has a sign, say ψ j (x j ) > 0 at every point of Π\ {v 1 , v N +1 }. By a well-known property of unrestricted minimizers of R in C(Π) ∩ H 1 (Π) = H 1 [0, L], see e.g. [1,2,7,18,19], the function ψ satisfies the Kirchhoff condition in (7), in particular But as a solution of the second order linear ODE ∂ 2 j ψ j + f (u j )ψ j = 0 on each edge k j , ψ 1 vanishes identically by uniqueness, as well as all the other ψ j do by connectedness of Π, by the Kirchhoff condition and by (3). Thus, by continuity, u must be constant, which leads to a contradiction. Now we consider the general Kirchhoff law (2) on the path. As on all trees due to its homogeneous character, it can be written in the form with positive constants c j on each edge, that leads to a self-adjoint resolvent by using a suitable scalar product and to the Rayleigh quotient with c = (c j ) N ×1 . Note that (11) is also the Kirchhoff law considered by Yanagida [21], as well as in [8] in order to reduce the eigenvalue asymptotics on trees to a consistent case. In fact, on the path Π, the form (11) is readily achieved by setting The same technique applies on arbitrary trees and shows that for equilibria the linearized stability under (2) and (11) are equivalent. Set As 0 is the minimal eigenvalue of the Laplacian − ∂ 2 j N ×1 , and as the only harmonic functions on a finite metric graph are the constant ones, an equilibrium u 0 of (1) on a tree leads to for all positive coefficients c.
Theorem 3.2. Let Π be a path. Then any stationary nonconstant solution u ∈ is unstable.
Proof. The assertion is true on an interval by Matano's result [18,21]. Thus, we can assume that N ≥ 2. Suppose that u ∈ C 2 K (Π) is a stable nonconstant stationary solution of (14) on Π. Distinguish the following three cases. If then u would be unstable by [9,Lemma 4.4] or Yanagida's Two Points Lemma [21], which is impossible. If then Π splits up at π j (x j ) into two paths having less than N edges and on which the restrictions of u would be unstable by recurrence, which is impossible.
Thus, we are lead to the final case Write (K) in the form (11) and set ψ j = c j ∂ j u j on each edge, where it solves ∂ 2 j ψ j + f (u j )ψ j = 0. By (11), the ψ j define a function If ∂ 1 ψ 1 (0) = 0, then ψ 1 vanishes identically on k 1 , and u has to be constant on k 1 . Omitting this edge, the restriction of u to the remaining edges has to be unstable by recurrence, which is impossible. The same argument applies to ∂ N ψ N ( N ) = 0. Thus, we can assume that Denoting the numerator of R ψ; u; c j We conclude that the edge Schrödinger operators lead to a Rayleigh quotient satisfying Clearly, ψ does not fulfill (16). But, if λ 0 = 0, then ψ is a minimizer of R ·; u; c j −1 and fulfills (16), in particular the Neumann condition at v 1 and v N +1 , which is impossible by (15). Thus, we can assume that there is η ∈ C 2 (Π) satisfying (16) such that This means that the zero solution is unstable for the edge operators (16). But all the norms Using the same argument as in [21,Lemma 2.3] in order to show that negative values of the Rayleigh quotient lead to instability, we conclude that u is unstable.
On a circuit ζ the derivatives of a nonconstant stationary solution u ∈ C 2 K (ζ) have to vanish at least twice. By cutting ζ at one of these points, we are led to a path to which the foregoing results can be applied. Thus we can state the Corollary 1. Let ζ be a circuit and u ∈ C 2 K (ζ) be a nonconstant stationary solution of the problem Then u is unstable.
Proof. Suppose that u is stable on ζ. By compactness, there is π j (x j ) = p ∈ ζ such that Cut the circuit at p into two boundary verticesṽ 1 andṽ N +1 and get a pathΠ whose quantities stemming from ζ will be denoted by a tilde. In particular,ũ ∈ C 2 K (Π) defines a nonconstant stationary solution of (19) onΠ with the distinguished propertyũ Clearly, C(ζ) can be identified with F. By Theorem 3.2,ũ is unstable onΠ. Thus, there is someε 0 > 0 such that for each δ > 0 there is an initial dataw 0 ∈ C(Π) But for each t > 0, z(·, t) belongs to F, andz is the solution of (19) onΓ with initial conditionz 0 = z(·, 0). This leads to the contradiction 4. Stability surgery. In this section we establish some basic facts and results about stationary stable or unstable classical solutions of the more general reaction -diffusion -problem under the hypotheses (6). In particular we compare the stability on a metric graph Γ with the one on a modified graphΓ obtained by cutting an edge at a zero of the derivative of a stationary solution. In [6,7], a general Osgood type L ∞ -estimate for nonlinear parabolic problems on metric graphs has been shown. For the reader's convenience and for the proofs of Theorems 4.2 and 6.1, we cite it here in the setting of the special case (5).
) be a solution of Problem (5) subject to the Osgood condition Then, the following estimate holds: The basic reduction tool for establishing instability is the following surgery technique.
is a stationary solution of (5) such that ∂ j u j (x j ) vanishes at p = π j (x j ) on some edge k j . Cut the graph Γ at p and get a new possibly non connected graphΓ on which u defines a stationary solutionũ of (5) onΓ that belongs to C 2 K (G). Here the Kirchhoff conditionsK extend the old one (K) by adding the Neumann condition at the new vertex or at the two new vertices. In the first case note that p = v i leads tod ij = 0.
Next, suppose thatw 0 ∈ ι(F ). Asũ belongs to ι(F ), and as w 0 −ũ ∞,G < δ, we can nevertheless choosez 0 ∈ C(G) ∩ ι(F ) with |supp(w 0 −z 0 )| sufficiently small such that z 0 −w 0 ∞,G < δ. Letz denote the solution of (5) onG with initial conditionz 0 and z be the solution of (5) on G with initial condition z 0 = ι −1 (z 0 ). Again, by uniqueness, z = ι −1 (z). Now we claim Since the coefficient of the linear term d j is bounded from above by b 1 , the Osgood type a priori estimate in Theorem 4.1 with b 2 = 0 yields This shows the claim ( * ). Finally, we conclude Thus, u is unstable on G.
Finally, in order to get rid of (20), i.e. for arbitrary nonlinearities f j ∈ C 1 (R), we modify them for the given stationary nonconstant solution u ∈ C 2 K (G) of (5) outside [−2M, 2M ] with M := u ∞,G as follows Thus, u would be a stable nonconstant solution of (28) with the nonlinearities f j iffũ would be a stable by the results shown first. This concludes the proof of the theorem.
As a first application of the surgery theorem 4.2 we generalize Yanagida's Two Points Lemma [9,21] known in the constant coefficients case under consistent Kirchhoff conditions to the more general problem (5).
is a stationary solution of (5) that is nonconstant on some edge k j . If there are two points on k j with 0 ≤ z 1 < z 2 ≤ j such that ∂ j u j (z 1 ) = ∂ j u j (z 2 ) = 0, then u is unstable. If, in addition, the nonlinearity f j is an odd function, then the same conclusion holds, if there are two points on k j with 0 ≤ z 1 < z 2 ≤ j such that u j (z 1 )∂ j u j (z 1 ) = 0 = u j (z 2 )∂ j u j (z 2 ).
Proof. In the first case, we cut the edge k j twice and obtain instability on the resulting sole edge as a stationary solution of a Neumann problem by a result by Matano [18]. Then Theorem 4.2 permits to conclude. In the second case, the assertion can be shown exactly in the same way as in the consistent and constant coefficient case shown in [9,Lemma 4.5].
As already pointed out in [9], the hereditary properties of the stability notion with respect to subgraphs are very bad. However, the edge doubling is a simple extension technique that permits to conclude for stability from a graph containing the original one. Choose any edge k j incident to v i and v h in Γ and copy all quantities and functions associated to k j on a new edge k N +1 incident to is a stationary solution of (5). LetΓ be the graph resulting from an edge doubling described above andũ ∈ C 2 K (G) be the corresponding extension of u toG by settingũ N +1 =ũ j = u j . Thenũ is stable oñ Γ if and only if u is stable on Γ.
Proof. Each w ∈ C 1 K (G) defines uniquelyw ∈ C 1 K (G). This leads to the continuous embedding Thus, identifying C(G) with F := w ∈ C(G)w N +1 =w j , the stability ofũ in C(G) implies its stability in the closed subspace F and thereby, the stability of u in C(G).
But for each t > 0, z(·, t) belongs to F, andz is the solution of (5) onΓ with initial conditionz 0 =z(·, 0). By construction, this leads to the contradiction A non trivial application of the edge doubling is given by the following example.
Example 4.1. Suppose that N = 2 = n, and that Γ consists in a loop ζ of length with ramification node {v 1 } = V (ζ) and in an edge k 3 joining v 1 and the boundary vertex v 2 . Then there is no stable nonconstant stationary solution of (5) on Γ. This will be shown as follows.
Suppose that u ∈ C 2 K (G) is a stable nonconstant stationary solution of (5). If u vanishes on the whole ζ or on the whole k 3 , then the problem is reduced to an interval under Neumann boundary conditions that admits only constant stable stationary solutions.
If v 1 is an extremum of u, then the problem on Γ splits into two interval problems. Thus, by Lemma 4.3 and compactness, the derivative ∂ j u j vanishes exactly once outside the ramification node v 1 on ζ in Γ, say at p ∈ ζ, since it cannot vanish in the interior of the edge k 3 . The point p cuts ζ into two edges, say k 1 and k 2 , on which u 1 and u 2 respectively are strictly monotone by Lemma 4.3. Moreover, u 1 and u 2 fulfill ∂ 1 u 1 (p) = ∂ 2 u 2 (p) = 0 and u 1 (p) = u 2 (p) . By unique solvability of the corresponding Cauchy problems, u 1 and u 2 coincide for 0 ≤ x j ≤ min { 1 , 2 }, where we have chosen d 11 = d 12 = 1. But, by continuity at v 1 u 1 (v 1 ) = u 2 (v 1 ) . Thus, if k 1 and k 2 were of different length, Rolle's Theorem and Lemma 4.3 would lead to instability of u. We conclude that k 1 and k 2 are of the same length . Choose d 13 = −1 and denote the boundary vertex by v 2 . Note that the conductivities on the loop at v 1 are identical, and the original Kirchhoff condition at v 1 reads Cutting ζ at p and omitting k 2 leads to a path Π of two edges given by k 1 and k 3 . It turns out that Γ is just the graphΠ with k 1 doubled with k 2 . Moreover, the restriction of u to Π, say w ∈ C 2 Kc (Π) and u =w, constitutes a stable stationary solution on Π and belongs to C 2 Kc (Π), where K c stands for the validity of the inconsistent Kirchhoff law (22) at v 1 and of the Neumann boundary condition at p and v 1 . Thus, by Theorem 3.2, w and u have to be constant, which leads to the desired contradiction.

Yanagida graphs. Recall the following result by E. Yanagida from 2001.
Theorem 5.1. ( [21]) If Γ is one of the five graphs in Figure 5, then the reactiondiffusion problem has no stable stationary nonconstant solution on G. This includes paths and circuits of arbitrary lengths of the same width d j > 0 on all their edges.
An important ingredient of Yanagida's proof was the self-adjoint character of the associated eigenvalue problem of the linearized problem. Note that Example 4.1 restricted to Problem (23), but without restriction on the d j , yields another exceptional graph on which no stable stationary nonconstant solution can exist. In this argument only Theorem 3.2, Example 4.1 and the classical Two Points Lemma [9,21] interfere. As for paths and circuits in Section 3, we can add the following exceptional graphs.  Figure 2. Yanagida's exceptional graphs.
does not admit any stable stationary nonconstant solution, if the graph Γ is a generalized Yanagida graph with arbitrary edge lengths of Type 3, 4, or 5, i.e.
(3) either Γ contains exactly one boundary vertex and exactly one essential ramification node, the latter being of valency 3, (4) or V b = ∅ and V ess = {v 1 } and γ 1 = 4, (5) or V b = ∅ and Γ contains exactly two essential ramification nodes, the latter being both of valency 3 and being joined by an edge or a viaduct.
Proof. Throughout, let us suppose that u ∈ C 2 K (G) is a stable nonconstant stationary solution of (24). If u is constant on some edge k j with d ij = 0 = d hj , then we can identify v i and v h and omit k j . Thus, we can assume that u is nonconstant on each edge of Γ.

Case (3)
By hypothesis, Γ is unicyclic. If some ∂ j u j vanishes at v 1 , the graph reduces to a sole path or splits into a circuit and a path. Working with C 1 -solutions, in particular at nodes of valency 2, we can assume w.l.o.g. that the circuit ζ is a loop of length and that the path joining v 1 and the boundary vertex v 2 is just an edge, say k 3 . Exactly as in Example 4.1 and using the notations given there, cut ζ at p ∈ ζ\ {v 1 } into two edges k 1 and k 2 with ∂ 1 u 1 (p) = ∂ 2 u 2 (p) = 0, u 1 (p) = u 2 (p) , and 1 = 2 = 2 . Then the Kirchhoff law (22) at v 1 reads Cutting ζ at p and omitting k 2 leads to a path Π of two edges given by k 1 and k 3 . Then the restrictionũ of u to Π is a stationary solution of the differential equations on the edges of Π and belonging to C 2 Kc (Π), where K c stands for the validity of the inconsistent Kirchhoff law (25) at v 1 and of the Neumann boundary condition at p and v 1 . Denoting by Σ the star formed by k 1 , k 2 and k 3 , respective identifying leads to the embeddings since each ϕ ∈ C 1 Kc (Π) extends uniquely to C 1 K (Σ) and C 1 K (Γ) by even extension with respect to π 1 (0) = p = π 2 (0) onto the remaining edge k 2 . This leads to stability of the restrictionũ of u to Π as follows. First, note that if w ∈ C 2,1 Kc (Π × [0, ∞)) is the solution on Π with initial condition w 0 ∈ C(Π), and ifw ∈ C 2,1 K (Γ × [0, ∞)) is the solution on Γ with initial conditionw 0 ∈ C(G) such thatw 0 = ι(w 0 ), then these solutions coincide by uniqueness of the corresponding flow, i.e.
Thus, the restriction of u to Π is stable there, which is impossible by Theorem 3.2. This achieves the proof in Case (3).

Case (4)
By Lemma 4.3, on each loop or circuit in Γ the derivative ∂ j u j vanishes exactly once outside the ramification node v 1 . As in Case (3), this allows the reduction to two pairs of edges of identical lengths that, in turn, lead to a restriction to a path of two edges even under a consistent Kirchhoff condition at the ramification node. As above, the two evolution flows are compatible with the extension-restriction procedure and permit to conclude with Lemma 3.1, or even with Matano's classical result on an interval.

Case (5)
Again, dealing with C 1 -solutions, we can assume w.l.o.g. that the two circuits ζ 1 and ζ 2 are loops of length 1 and 2 , respectively, and that the viaduct between the nodes v 1 and v 2 of Γ is just an edge denoted by k 3 . Assume that v i ∈ V (ζ i ).
If ∂ j u j vanishes at some point on k 3 , then we are led to two graphs of Case 3, or one of this type and a loop. Thus, it remains to consider the case where the derivative ∂ j u j vanishes exactly once on each ζ i outside v i . As in Case (3), this allows the reduction to two pairs of edges of identical lengths that, in turn, lead to a restriction to a path of 3 edges under an inconsistent Kirchhoff condition at v 1 and v 2 as in (25). As above, the two evolution flows are compatible and permit to conclude with Theorem 3.2. Figure 5 presents some graphs fulfilling the conditions of Theorem 5.2, that are not in Yanagida's list. Note that the embedding (26) yields compatibility of solutions for the extension-reduction-procedure related to appropriate subgraphs. However, in general graphs, such embeddings compatible with the different involved flows do not seem to be available. They would strongly simplify subgraph reduction techniques in showing instability. Therefore, the surgery techniques from Section 4 will be applied for the general case, rather than the ones above. 6. Proof of the main result. In this section we shall use the notation for the outer normal derivative of a function u on the edge k j at the vertex v i ∈ k j . First, we consider Problem (1) on a tree T , i.e.
under Condition (4). Theorem 3.2 settles the case of a tree without essential ramification nodes and forms part of the following Theorem 6.1. On any finite metric tree T there is no stable stationary nonconstant solution of Problem (28).
Proof. Let u ∈ C 2 K (T ) be a stable stationary nonconstant solution of (28). For a sole edge or a path this is impossible due to Theorem 3.2. Thus we can suppose that N > 2 and that #V ess ≥ 1, and reason by recurrence on N . Modifying f for the given solution u ∈ C 2 K (T ) of (28) outside [−2M, 2M ] with M := u ∞,T as in (21) for the proof of Theorem 4.2, we can assume w.l.o.g. that with some constant b 1 ≥ 0. Moreover, recall that under a dissipative Kirchhoff condition, i.e. all c ij > 0, as the third condition in (28), differentiable functions at an extremum in a vertex behave like at an interior point, and all incident derivatives have to vanish there.
If ∂ j u j (x j ) = 0 for some x j ∈ [0, j ] with k j ∩ V b = ∅, then T splits up at π j (x j ) into two trees having less than N edges and on which the restrictions of u are stable. This is impossible by recurrence.
Thus, we conclude that ∂ j u j (x j ) = 0 except at the boundary vertices V b at which clearly ∂ j u j (v i ) = 0. It follows that on each edge k j , the function u j is strictly monotone.
Choose any essential ramification node v i . Then there are at least two incident edges, say k 1 and k 2 , satisfying Remove k 1 from T and get two disjoint subtrees of T . LetT denote the one containing v i and setc Then the restriction of u toT belongs to C 2 K (T ) with the dissipative Kirchhoff law and constitutes a stationary nonconstant solution of (28) onT that has to be unstable by recurrence. Thus, there is some ε 0 > 0 such that for each δ > 0 there is an initial dataw 0 ∈ C(T ) with w 0 − u ∞,T < δ and w(·, t 0 ) − u ∞,T ≥ ε 0 for some t 0 > 0 wherew denotes the solution of (28) with initial conditionw 0 onT .
Next, we modify and extendw 0 to a function w 0 on T as follows. Choose δ 0 sufficiently small such that for all 0 < δ ≤ δ 0 ≤ ε0 4 e −b1t0 , there exists w 0 ∈ C(T ) that coincides with u in a small neighborhood of v i and fulfills the constraints Then define w 0 outsideT on T simply by u. Evidently, Moreover, for the solution w of (28) with the initial data w 0 on T we claim Since the coefficient of the linear term z j is bounded from above by b 1 , the Osgood type a priori estimate in Theorem 4.1 with b 2 = 0 yields z(·, t 0 ) ∞,T ≤ e b1t0 max T |z(·, 0)| = e b1t0 w 0 −w 0 ∞,T < ε 0 2 .

JOACHIM VON BELOW AND JOSÉ A. LUBARY
This shows the claim ( * ). By stability of u on T , there is a δ ∈ (0, δ 0 ] such that the initial data w 0 from above with w 0 − u ∞,T = w 0 − u ∞,T < δ leads to the solution satisfying w(·, t) − u ∞,T < ε 0 2 for all t > 0. OnT evaluated at t 0 > 0, this leads to which is absurd.
In order to achieve the proof of the general case we need a technical combinatorial lemma for graphs with circuits. Lemma 6.2. Let Γ be a finite graph that contains circuits of lengths at least 2. Let the set of real numbers {∆ ij 1 ≤ i ≤ n, 1 ≤ j ≤ N } satisfy the following properties: Then there exists an edge k j with ramification nodes v i , v h ∈ k j or a viaduct π with endpoints v i , v h such that there are two edges k r and k s with ∆ ij ∆ is > 0 and ∆ hj ∆ hr > 0.
Proof. Note first that, by (c) and (d), each circuit ζ in Γ contains a pair of edges k j , k s incident to v i ∈ V (ζ) such that ∆ ij > 0 and ∆ is > 0 or a pair of edges k l , k t incident to v m ∈ V (ζ) such that ∆ ml < 0 and ∆ mt < 0. Let ζ be a circuit of length m with V (ζ) = {v 1 , . . . , v m } ordered by the relations d ii = −1 and d i+1,i = 1 with indices to be taken mod m. Suppose that W.l.o.g. assume that ∆ 11 > 0 and ∆ 1m > 0 and that each viaduct is replaced by an edge of corresponding length. Then each By (c), ∆ 21 < 0. If ∆ 22 < 0 or ∆ 2j < 0 for some incident edge outside ζ, then the assertion is shown. Thus, we can assume ∆ 22 > 0, ∆ 2j > 0, and ∆ 32 < 0.
It follows recursively, that for ∆ i,i−1 < 0, if ∆ ii < 0 or ∆ ij < 0 for some incident edge outside ζ, then the assertion is shown with v i and v i−1 . Thus, we can assume ∆ ii > 0, ∆ ij > 0, and ∆ i+1,i < 0 with some incident edge k j outside ζ. If no pair v i and v i−1 for i ≤ m − 1 has been found yet as asserted, then, finally, v m−1 and v m (as well as v m and v 1 ) will fulfill the requirements since then Proof. W.l.o.g. by introducing artificial nodes with Kirchhoff conditions leading to continuous differentiability, we can assume that Γ is simple, i.e. Γ has neither loops, nor multiple edges. We shall reason by recurrence on d := corank(Γ). Recall that the circuit space Π(Γ) of the graph Γ is defined by and satisfies Π(Γ) = ker D(Γ), see e.g. [12]. Moreover that amounts to N − n + 1 for connected graphs. For d = 0, the assertion is true by Theorem 6.1. Thus, we can suppose d ≥ 1.
Let u ∈ C 2 K (G) be a stable stationary nonconstant solution of (1). Let k j be an edge such that ∂ j u j (x j ) = 0 for some x j ∈ [0, j ] and set p = π j (x j ). First, consider the case that 1. If k j is incident to a boundary vertex, then Lemma 4.3 permits to conclude. 2. If k j is a bridge that is not incident to V b , then cutting at p leads to two disjoint metric subgraphsΓ 1 andΓ 2 of G. If one of them, sayΓ 1 , were a tree, then u would have to be unstable by Theorem 4.2 and Theorem 6.1. If both of them contain circuits, then clearly By recurrence, the restrictions of u toΓ 1 andΓ 2 are unstable, and so is u by Theorem 4.2. 3. If k j belongs to a circuit, then cutting at p leads to a graphΓ of corank d − 1.
By recurrence the restriction of u toΓ must be unstable, and so does u by Theorem 4.2. Thus, it remains to show the assertion in the case

Use notation (27). At the endpoints
or ∆ ij ∆ hl < 0, respectively. Thus, w.l.o.g. we can consider any viaduct as an edge in the remaining reasoning. The set of ∆ ij clearly satisfies the hypotheses (a)-(c) of Lemma 6.2, while Condition (d) is fulfilled by (6), by the continuity requirement at the nodes and by the strictly monotone character of each u j . Thus, Lemma 6.2 guarantees the existence of an edge k 1 with ramification nodes v i , v h ∈ k 1 such that there are two edges k r and k s with ∆ i1 ∆ ir > 0 and ∆ h1 ∆ hs > 0, respectively. Then introduce the modified conductivities bỹ Finally, omit the edge k 1 in Γ and proceed on the resulting and possibly non connected graphΓ as in the proof of Theorem 6.1. This is possible by the local character of the modifications applied there in the vicinity of k 1 .
7. Common Hamiltonian edge system. The stationary case of the fully autonomous consistent parabolic problem leads to the same first order system (33) defined by f and = ∂ j on each edge, i.e.
(1, 0) that satisfy u L ∞ (G) > 1. But in all the cases, by Lemma 2.2, a nonconstant stationary solution u satisfies For a Lyapunov-energy-calculus we introduce ) be a solution of (1). (a) Along u the energy decreases: and In particular, if u is an equilibrium A, then E(u) = −LH(u, 0).
Proof. As for (a), we can follow a standard density argument using the Kirchhoff and the continuity condition: As for (b), both assertions follow readily with the definitions and from Proposition 1.
In order to apply Lasalle's principle, we have to impose an additional condition to f . E.g. under the hypothesis we obtain with M := max R F + < ∞ that This enables the application of Lasalle's Principle [4] in order to conclude the following Corollary 2. Under Condition (36) the solutions u ∈ C(G × [0, ∞)) ∩ C 2,1 K (G × (0, ∞)) of (32) tend to stationary solutions as t → ∞ with respect to · L ∞ (G) , since their ω-limits belong to the set of functions satisfyingĖ(u) = 0.
We apply the preceding results to the attractivity properties of the equilibria for a nonlinearity f subject to the following conditions.
They clearly include the case of a cubic f ( Moreover, Condition (36) is satisfied with and Corollary 2 applies. Now we can state the following results about the flow defined by Problem (32) subject to Condition (37).
The equilibrium A is a local attractor, whose domain of attraction satisfies The equilibrium C is a local attractor, whose domain of attraction satisfies Proof. The claimed inclusions in (a) and (b) follow with [9, Theorem 4.1]. By continuity, Corollary 2 ensures that the solutions u ∈ C(G × [0, ∞)) ∩ C 2,1 K (G × (0, ∞)) of (32) with initial data u 0 belonging to D(A) or to D(C) satisfy lim t→∞ G u(·, t)dx = A G dx = LA and lim t→∞ G u(·, t)dx = LC, respectively. It follows that D(A) and D(C) belong to the sets in the middle. But according to Lemma 7.1, a nonconstant stationary solution w satisfies Thus, for u 0 belonging to one of the sets in the middle, its solution has to be attracted by the equilibrium A or C, respectively.
The assertion (c) follows readily by Lemma 7.1 and (a) and (b). As for (d), For the balanced cubic f (u) = u 3 − u the equilibrium 0 is a global attractor, see [9, p.180], while for f (u) = u − u 3 it is not a repeller, since its basin of attraction is given by the set of continuous initial conditions u 0 , whose solutions satisfy lim t→∞ E(u(·, t)) = 0, see [9,Prop. 6.9]. Theorem 7.3 applies in particular to the balanced cubic f (u) = αu(A − u)(u − A) that defines the nonlinearity of the Schlögl system. For the latter one it has been shown in [15] that the equilibrium 0 for the Schlögl system on a sufficiently small interval can by L 2 -stabilized exponentially fast by a suitable Robin boundary feedback condition. The open question arises whether an analogous result holds on general metric graphs too. 8. The non autonomous case and other transition conditions. The smallest example of the existence of a stable nonconstant stationary solution in presence of reaction terms depending on the edges is the following one.
Example 8.1. Let Γ be the path of length 2 with 1 = 2 = 1 and the orientation and labeling given by (9). Define w ∈ C 2 K (G) by Then ∂ 2 1 w 1 +1 = 0 and ∂ 2 2 w 2 −1 = 0 in [0, 1], and u is stable with respect to the flow generated by the edge evolution equations ∂ t u j = ∂ 2 j u j −(−1) j in C 0 (G) (or L 2 (G)). This follows from the fact that for any solution u ∈ C(G × [0, ∞)) ∩ C 2,1 K (G × (0, ∞)), the difference δ = u − w solves the heat equation ∂ t δ j = ∂ 2 j δ j on each edge k j . But the minimal eigenvalue of the Laplacian − ∂ 2 j N ×1 under (2) and (3) is 0. Thus, eigenfunction expansion and Dirichlet's Theorem yield Example 8.3. If the nonlinearity depends on the edges and on x j , but not on u, then either there is no stationary solution or there is a unique stable one, that can be nonconstant. In detail and generalizing Example 8.1, for given f j ∈ C[0, j ] we consider the inhomogeneous heat flow problem K (G × (0, ∞)), ∂ t u j = ∂ 2 j u j + f j (x j ) on k j for 1 ≤ j ≤ N, If G f (x) dx = 0, then there is no stationary solution w ∈ C 2 K (G) of (39), since this would lead to But if f fulfills G f (x) dx = 0, then a unique stationary solution w ∈ C 2 K (G) of (39) can be obtained as follows. Introduce Using (3) and (K), it readily follows as in [5,7,10,16,17], that there exist unique coefficients b j and c j for 1 ≤ j ≤ N such that w ∈ C 2 K (G) defined by is the unique stationary solution of (39). As in Example 8.1, w is seen to be stable, since for every solution u of (39), the difference u − w solves the heat equation in C(G × [0, ∞)) ∩ C 2,1 K (G × (0, ∞)). Example 8.4. If the diffusion coefficients are allowed to be different, then again stable nonconstant stationary solutions can occur. The example follows a refinement of Matano's type of counterexamples for non convex domains established by Cònsul and Solà-Morales [13]. Consider the path Π with 3 edges using the numbering (9) and choosing the edge lengths and diffusion coefficients a j to be 1 = 3 = 1, 2 = δ > 0, a 1 = a 3 = 1, a 2 = ε > 0 with δ and ε sufficiently small to be determined later. As common nonlinearity we choose f (u) = u − u 3 . Using the double well potential G(u) = 1 4 (1 − u 2 ) 2 and the modified energy it can be shown that for ε = δ 2 , there exists a minimizer w of E in u ∈ C(Π) ∩ H 1 (Π) that is close to −1 on k 1 and close to 1 on k 3 . Moreover, w is stable.
then the stability results change dramatically. E.g., there are no exceptional graphs, since for any finite metric graph, there is a suitable nonlinearity f such that there is a stable nonconstant stationary solution (u j ) N ×1 governed by the edge equations ∂ 2 j u j + f (u j ) = 0. We refer to [10] for the details. Remark 8.2. As already pointed out above, node transition conditions different from the ones given by (2) and (3) can allow stable nonconstant stationary solutions. In particular, consider the limit problem of a parabolic problem on Γ depending on a parameter and allowing only constant stable stationary solutions. If the limit problem changes the type of its transition conditions or even the type of its edge differential equations, then stable nonconstant stationary solutions can occur in the limit. An example is given by Problem (1) with (2) replaced by the dynamical Kirchhoff condition j d ij c ij ∂ j u j (v i , ·) + σ∂ t u(v i , ·) = 0, see [11]. Letting σ tend to ∞ leads to a parabolic problem with a certain Dirichlet condition at the nodes, that can allow stable nonconstant stationary solutions using a similar argument as in 8.2.
As the stationary solutions of the parabolic and the corresponding hyperbolic problem are the same when keeping the same node transition conditions, the interesting question arises, whether stability properties can be carried over from one case to the other. More generally, if the hyperbolic problem is approximated by parabolic ones with established stability criteria, e.g. by relaxation, is it possible to conclude stability criteria for the hyperbolic case from those in the parabolic case? A precise answer to this question is certainly of big interest and could apply e.g. to gas networks considered in [14], where the existence of nonconstant stationary solutions has been established for certain metric graphs.