Gaussian estimates for a heat equation on a network

We consider a diffusion problem on a network on whose nodes we impose Dirichlet and generalized, non-local Kirchhoff-type conditions. We prove well-posedness of the associated initial value problem, and we exploit the theory of sub-Markovian and ultracontractive semigroups in order to obtain upper Gaussian estimates for the integral kernel. We conclude that the same diffusion problem is governed by an analytic semigroup acting on all $L^p$-type spaces as well as on suitable spaces of continuous functions. Stability and spectral issues are also discussed. As an application we discuss a system of semilinear equations on a network related to potential transmission problems arising in neurobiology.


Introduction
Evolution equations taking place in networks or, more generally, in ramified structures have been first considered in pioneering articles by K. Ruedenberg and C. Scherr back in the 1950s, cf. [37], and, at a more mathematical level, in a series of papers by R. Mills and E. Montroll and by G. Lumer in the 1970s, cf. [30]- [29] and [25]- [26], respectively. Shortly afterwards, F. Ali Mehmeti, J. von Below, S. Nicaise, and J.P. Roth among others began a systematical study of properties of elliptic operators acting on spaces of functions over networks, cf. e.g. the monographs [35], [2], [24], and references therein. Ever since, such problems have aroused broad interest among mathematicians working on partial differential equations, control, and spectral theory -as well as among theoretical physicists interested in scattering theory of guided waves, photonic crystals, and quantum wires, resulting in a literature so vast that it can by no means be summarized here.
Throughout this paper we consider a finite, unitarily parametrized, connected network whose structure is given by a suitable graph. On it we study a general diffusion equation. Adopting a setting which is standard in literature, the node conditions impose continuity and Kirchhoff-type transmission laws in ramification vertices. However, we extend known results by allowing more general, non-local node conditions, cf. Section 2 below. Roughly speaking, in each node v i of the network we allow an absorption that does not only depend on the value of the function in v i itself, but also on other nodes v h . Using the arguments presented in [11], one can promptly obtain well-posedness of such a general parabolic network problem; we are thus more interested in qualitative properties. In some sense, our results complement those obtained in [19] and [20], where even more general node conditions are allowed, and where positivity of the semigroups is also discussed, but where the quantum physical viewpoint was motivated to mainly discuss those conditions leading to self-adjointness.
In this paper we pursue an approach to parabolic equations on networks based on the theory of sesquilinear forms and associated sub-Markovian semigroups. Following the approach presented in previous works (see [11], [2,Chapt. 2], and [21]), we draw some conclusions about several issues, including L ∞ -contractivity of the semigroup governing the problem, its L 2 − L ∞ -stability and its analyticity in suitable spaces of continuous as well as of L 1 functions over the graph. The key arguments are upper estimates for the integral kernel of the generated semigroup (which, roughly speaking, yields the Green function of the problem). The theory of such Gaussian estimates has become a mature one that proves extremely powerful when applied to diffusion problems on domains -and, as a matter of fact, on networks, too; we refer to [15], [3], and [36] for an introduction to this subject.
While the integral kernel for the diffusion problem on a network has already been explicitly computed in [34] in the special case of constant coefficients for the heat operator, Gaussian estimates are, to our knowledge, new in the case of variable coefficients, even in the case of local nodal conditions. Observe also that Gaussian estimates for the semigroup that governs the discrete diffusion problem on a graph have recently been proved in [23], cf. also [8].

General framework
We consider a finite connected network, represented by a finite graph G with m edges e 1 , . . . , e m and n vertices v 1 , . . . , v n . We normalize and parametrize the edges on the interval [0, 1]. The structure of the network is given by the n × m matrix Φ + := (φ + ij ) and Φ − := (φ − ij ) defined by For the sake of simplicity, we denote the value of the functions c j (·) and u j (t, ·) at 0 or 1 by c j (v i ) and u j (t, v i ), if e j (0) = v i or e j (1) = v i , respectively. With an abuse of notation, we also set u ′ j (t, v i ) = c j (v i ) := 0 whenever j / ∈ Γ(v i ). In the literature networks are usually considered where Dirichlet conditions are imposed on n 0 boundary vertices (i.e., vertices of degree 1). Since no process takes place in such boundary vertices, we may and do identify all of them. Thus, we instead consider an equivalent diffusion equation on a rearranged graph where these n 0 nodes of degree 1 are replaced by only one node of degree n 0 , on which a Dirichlet condition is imposed. Without loss of generality we assume that this node is v n .
For t ≥ 0 we are now in the position to consider the network diffusion problem x ∈ (0, 1), j = 1, . . . , m.
Throughout this paper we assume that the coefficients c j satisfy 0 < c j ∈ C 1 [0, 1], j = 1, . . . , m, while for the time being the (n − 1) × (n − 1) matrix B := (b ih ) 1≤i,h≤n−1 is arbitrary, i.e., B ∈ M n−1 ( ). The third equation above is a generalized, non-local Kirchhoff-type law on the first n − 1 nodes, while the fourth one prescribes a Dirichlet condition in v n . We introduce the n × m matrices Φ + w := (ω + ij ) and Φ − w := (ω − ij ) defined by otherwise. In the following we will repeatedly and without further notice write the functions u j in vector form, i.e., With these notations, one can directly check that the second, third and fourth equations of our network diffusion problem can be rewritten as where we have introduced the n × n matrices Remark 2.1. Inspired by the non-local boundary conditions introduced above, and motivated by the theory of ephaptic coupling of myelinated nerve fibers (see [38,Chapt. 8]) one may also consider an ever more general parabolic system over the network, where for t ≥ 0 the first equation of the problem is replaced byu Under suitable assumptions on the diffusion coefficients c ij one can study well-posedness and qualitative properties of such a system. However, this introduces new technical difficulties, as for instance one sees that usual Kirchhoff-conditions do not ensure dissipativity of such a system any more, and one has to formulate appropriate, more general conditions in the nodes. We will discuss this kind of problems in the forthcoming paper [14].
It is already well-known that the above network diffusion problem is well-posed: in fact, this has been shown in a Hilbert space setting in [11] for the case of variable coefficients and diagonal B, cf. also the references therein for earlier results on less general cases. We also remark that, at least in the case of c 1 = . . . = c m ≡ 1 and b ih = 0, i, h = 1, . . . , n. S. Nicaise has derived in [34] an explicit formula for the solution, thus showing the well-posedness of the problem in all L p -type spaces as well as on suitable spaces of continuous functions (see also [33]). Our first goal is to to establish a meaningful L p -theory for the general case of variable diffusion coefficients and non-local node conditions. To this aim, we need the following.
With this notation one sees that the following holds.
The mapping U is one-to-one from X p := (L p (0, 1)) m onto L p (0, m) for all p ∈ [1, ∞], and in fact it is an isometry if we endow (L p (0, 1)) m with the canonical l p -norm, i.e., In the following we will hence regard X p as an L p -space over a finite measure space, so that X p ֒→ X q for all 1 ≤ q ≤ p ≤ ∞. Moreover, each X p is a Banach lattice, and its positive cone can be identified with the positive cone of L p (0, m).

Basic results
We are now in the position to consider an abstract reformulation of our diffusion problem. First we consider the (complex) Hilbert space X 2 = L 2 (0, 1) m endowed with the natural inner product On X 2 we define an operator We can finally rewrite the concrete network diffusion problem as an abstract Cauchy problem on X 2 . In order to discuss the semigroup generator property of A it is convenient to use a variational method.
Recall that we are assuming the network to be connected throughout the paper. This is crucial for the proof of the following.
is densely and compactly embedded in X 2 . It becomes a Hilbert space when equipped with the inner product Proof. It is well-known that V 0 is densely and compactly imbedded in X 2 : this follows from the inclusions (C ∞ c (0, 1)) m ⊂ V 0 ⊂ (H 1 (0, 1)) m and the Rellich-Khondrakov Theorem.
We are going to show that the inner product defined in (3.3) is equivalent to that induced by the Hilbert space (H 1 (0, 1)) m , i.e., to Let e j be a general edge. By the connectedness of G we can find a set of edges e j 1 , . . . , e jr linking any x ∈ e j with v n . More precisely, there exist j 1 , . . . , j r ∈ {1, . . . , m} such that • φ − nj 1 = 1, i.e., v n is the head of e j 1 ; • for all h = 1, . . . , r there exists a vertex v i h+1 such that φ + i h+1 j h = 1 = φ − i h+1 j h+1 , i.e, v i h+1 is the tail of e j h and the head of e j h+1 ; • e jr = e j .
Then, for every f ∈ V and at any x ∈ (0, 1) one has f jr (x) = d f i r−1 + x 0 f ′ jr (s)ds, and therefore due to the Dirichlet condition satisfied at v n by all f ∈ V . We conclude that Having proved such a Poincaré-type inequality, the claim follows directly.
The following lemma extends known results (see e.g. [1], [34], and [11]) to the case of non-local node conditions. Thus, for the sake of self-containedness we do not omit the proofs, although basic elements and techniques involved are essentially well-known.
on the Hilbert space X 2 . Then a enjoys the following properties: • a is X 2 -elliptic, i.e.,: there exist ω ∈ Ê, α > 0 such that • a is continuous, i.e., there exists M > 0 such that • a is symmetric, i.e., if and only if B is self-adjoint; • a is coercive, i.e., it is X 2 -elliptic with ω = 0, if and only if it is accretive, i.e., if and only if B is dissipative.
Proof. First of all, we show that a is X 2 -elliptic. We observe that the leading term in the form a, i.e., is clearly X 2 -elliptic and continuous. Furthermore, it follows from the Gagliardo-Nirenberg inequality for all u ∈ H 1 (0, 1), see e.g. [12,Théo. VIII.7], that the space is embedded in an interpolation space of order 1 2 between V 0 and X 2 , and obviously  b and this is the case if and only if B is self-adjoint. Let finally a be coercive. Then it is clearly accretive. If a is accretive, then a direct computation shows that holds for all f ∈ V 0 . Due to the arbitrarity of the nodal values d f 2 , . . . , d f n of f ∈ V 0 , one sees that this already implies that B is dissipative. Finally, if B is dissipative, then there holds By Lemma 3.1, we have thus obtained that Rea(f, f ) ≥ α f 2 V 0 for some α > 0. This concludes the proof. Corollary 3.3. The operator associated with a is densely defined, sectorial, and resolvent compact, hence it generates a strongly continuous, analytic, compact semigroup (T 2 (t)) t≥0 on X 2 . If moreover B is dissipative (resp., self-adjoint), then (T 2 (t)) t≥0 is contractive and uniformly exponentially stable (resp., self-adjoint).
Proof. It follows from Proposition 3.2 and [36, Prop. 1.51, Thm. 1.52] that the operator associated with a is densely defined, sectorial, and resolvent compact.
Let B be dissipative and take f in the null space of the operator associated with a. Then by the proof of the above lemma there exists c > 0 such that This means that f j is constant for all j = 1, . . . , m, hence f is constant on the whole network, due to the continuity condition in the nodes. In particular, f ≡ d f n = 0, hence the operator associated with a is one-to-one.
In the following we are able to identify the operator associated with a. The following result is already known in the literature in the special case where the matrix B is diagonal, see e.g. [2, Chapt. 2]. Proof. Denote by (C, D(C)) the operator associated with a, which by definition is given by Using the incidence matrix Φ = Φ + − Φ − and recalling that d u n = 0 as u ∈ V 0 , we can write which makes sense because Af ∈ X 2 . The proof of the inclusion A ⊂ C is completed.
To check the converse inclusion take f ∈ D(C). By definition, there exists g ∈ X 2 such that hence in particular for all functions of the form From this follows that (3.4) in fact implies By definition of weak derivative this means that Moreover, integrating by parts as in the proof of the first inclusion we see that if (3.4) holds for all u ∈ V 0 , then there also holds holds for all h ∈ V 0 . This implies that Af = −g, and the proof is complete.
By the above results, the operator A generates on X 2 a semigroup (T 2 (t)) t≥0 : thus, the abstract Cauchy problem (ACP) (and hence the concrete diffusion problem on the network) is well-posed in X 2 . We can characterize several features of (T 2 (t)) t≥0 by those of (e tB ) t≥0 , and hence of the scalar matrix B: we are going to show that (T 2 (t)) t≥0 is real, positive, and X ∞ -contractive (i.e., the unit ball of X ∞ is invariant under T 2 (t) for all t ≥ 0), respectively, if and only if the semigroup (e tB ) t≥0 generated by B is real, positive, and ℓ ∞ -contractive (i.e., once endowed n−1 with the equivalent ∞-norm, e tB leaves the unit ball of n−1 invariant for all t ≥ 0), respectively.
Theorem 3.5. The semigroup (T 2 (t)) t≥0 on X 2 associated with a enjoys the following properties: • it is real if and only if the matrix B has real entries; • it is positive if and only if the matrix B has real entries that are positive off-diagonal, If moreover B is dissipative, then (T 2 (t)) t≥0 is X ∞ -contractive if and only if where sign denotes the generalized (complex-valued) function defined by as well as By definition, the subspace V 0 contains exactly those functions on the network that are continuous in the vertices and vanish in the vertex v n . Then for every This holds for all f ∈ V 0 if and only if b ih ∈ Ê.
Moreover, if f ∈ V 0 , then ((Ref ) + ) j = (Re(f j )) + , 1 ≤ j ≤ m, and one sees as above that (Ref ) + ∈ V 0 . In particular, for all i = 1, . . . , n − 1 there holds Finally, we discuss the property of X ∞ -contractivity for the semigroup associated to a. Thus, take f ∈ V 0 . Then (1 ∧ |f |)signf ∈ H 1 (0, 1) m and, again, the continuity of f in the vertices imposes the same property to the function (1 ∧ |f |)signf , and in fact where b denotes the sesquilinear, accretive form on n−1 associated to the matrix B. Since the nodal values d f 1 , . . . , d f n−1 are arbitrary, by [36,Thm. 2.15] (which of course also applies to the accretive form b) one sees that the property of X ∞ -contractivity for (T 2 (t)) t≥0 is equivalent to that of ℓ ∞ -contractivity for the semigroup (e tB ) t≥0 generated by B on n−1 . Now the claim follows by Lemma 6.1.
We recall that if (T (t)) t≥0 and (S(t)) t≥0 are semigroups on a Banach lattice X, then (T (t)) t≥0 is said to dominate (S(t)) t≥0 in the sense of positive semigroups if Proposition 3.6. Let B,B be (n − 1) × (n − 1) dissipative matrices and denote by (e tB ) t≥0 and (e tB ) t≥0 the semigroups they generate on n−1 . Assume that B is real and positive off-diagonal, so that (e tB ) t≥0 is positive. Denote by a B , aB the coercive form a with coefficients given by B andB, respectively, and by (T B (t)) t≥0 , (TB(t)) t≥0 the associated semigroups on X 2 . Then (T B (t)) t≥0 dominates (TB(t)) t≥0 in the sense of positive semigroups if and only if (e tB ) t≥0 dominates (e tB ) t≥0 in the sense of positive semigroups.
Proof. Observe that V 0 , the domain of both a B and aB is an ideal of itself by [36,Prop. 2.20]. A direct computation shows that Due to the arbitrarity of the nodal values of f, g ∈ V 0 , such a condition is satisfied if and only if Thus, applying [36,Thm. 2.21] to the forms a B , aB as well as to the forms associated to the matrices B,B yields the claim.
It has been shown in [21, § 5] that the positive semigroup governing a diffusion problem on a network without Dirichlet conditions on any node is irreducible if and only if G is connected. This is no more true in the setting considered in this paper, unless we replace the notion of connectedness by a stronger one. (1) If the positive semigroup (T 2 (t)) t≥0 is not irreducible, then G is the union of two non-trivial subgraphs G 1 , G 2 ⊂ G containing vertices v 1 , . . . , v n 0 and v n 0 +1 , . . . , v n−1 , respectively, and such that G 1 ∩ G 2 ⊂ {v n }. (2) The converse also holds, if further B is assumed to be block-diagonal, i.e., of the form f ½G ∈ V 0 and then eitherG or its complement has zero measure.
In particular, let (T 2 (t)) t≥0 be irreducible and let f ∈ V 0 be a function that agrees with the constant 1 on all edges ofG that are not incident to v n , and is smooth elsewhere. Then f ½G is of class H 1 on each edge, continuous on each node of G 1 and vanishes in v n . Since f ½G also vanishes on all nodes of G 2 , it follows that f ½G ∈ V 0 . We deduce that ifG contains an interior point x of some edge e j , j ∈ Γ(v n ), then the whole e j belongs toG (otherwise f ½G would be discontinuous at x, and in particular not of class H 1 .) 1) Let (T (t)) t≥0 be not irreducible. We want to show that G is the union of nontrivial subgraphs whose intersection is {v n } or, in other words, that for all pair of points x, y ∈ G every path p(x, y) connecting them contains v n . Since the semigroup is not irreducible, there existsG ⊂ G such that µ(G) = 0 = µ(G \G) and such that (3.7) holds. As remarked before,G and G \G can thus only contain whole edges: let thus e j 0 , e ℓ 0 be edges contained inG and G \G, respectively. Assume now that then there exists a path p ⊂ G that connects some interior point of e j 0 to some interior point of e ℓ 0 and such that v n / ∈ p. However, by a continuity argument the function f ½G would be constant 1 along the path p, a contradiction. Here we have denoted by f a smooth function in V 0 that agrees with the constant 1 everywhere beside on the edged incident to v n .
2) TakeG = G 1 and let f ∈ V 0 . Then f ½ G 1 is a function that equals f on the edges of G 1 : by definition, f ½ G 1 is continuous in the vertices of G 1 and it vanishes in v n . Furthermore, f ½ G 1 vanishes on the edges of G 2 and all the vertices adjacent to them, thus in particular it is continuous in the vertices of G 2 , too.
Summing up, f ½ G 1 ∈ V 0 and moreover However, G 1 is not a set of measure zero.
3) LetG ⊂ G with µ(G) = 0, and assume that (3.7) holds. We are going to show that µ(G \G) = 0. In fact, let µ(G \G) = 0, i.e., let G \G contain (at least) one whole interval. We can thus assume that there exist nodes v i 0 ∈G and v h 0 ∈ G \G. Let us now consider some function a contradiction to (3.7).

Extrapolating semigroups, ultracontractivity, and Gaussian estimates
Let the matrix B and hence its adjoint B * be dissipative. One can easily see that the adjoint of the form a is given by the form a * defined by By Theorem 3.5 the semigroup associated to a * is X ∞ -contractive if and only if By duality, this is the case if and only if (T 2 (t)) t≥0 is X 1 -contractive. By interpolation we thus conclude that (T 2 (t)) t≥0 is X p -contractive for all p ∈ [1, ∞] if and only if both (3.6) and (4.1) are satisfied. We thus obtain the following.
Finally, for p ∈ (1, ∞) the spectrum of A p is independent of p, where A p denotes the generator of (T p (t)) t≥0 , and (T p (t)) t≥0 is uniformly exponentially stable with common growth bound ω(T p ) given by s(A).
Proof. We are in the position to apply the results summarized, e.g., in [4, § 7.2] and deduce the existence of an extrapolation semigroup (T p (t)) t≥0 on X p , 1 ≤ p ≤ ∞. A list of the all properties of (T 2 (t)) t≥0 inherited by (T p (t)) t≥0 can be found in [4, § 7.2.2]. This yields all the claimed properties up to exponential stability. To check this, we combine the uniform exponential stability of the semigroup associated with a, cf. Corollary 3.3, and the p-independence of the spectrum of the analytic semigroups (T p (t)) t≥0 , p ∈ (1, ∞). Hence, the growth bound of (T p (t)) t≥0 agrees with the spectral bound s(A p ) = s(A 2 ) = s(A) < 0. In other words, we can say that the more edges belong to the network, the slower is the heat dissipation. We have shown that, if B is dissipative and satisfies (3.6)-(4.1), then the semigroup (T p (t)) t≥0 is contractive on X p , p ∈ [1, ∞]. In fact, we can characterize a stronger property. (1) If B satisfies (3.6), then the semigroup (T 2 (t)) t≥0 on X 2 associated with a is ultracontractive. In particular, it satisfies the estimate for some constant M .

holds.
Proof. By [36, Thm. 6.3] it suffices to show that there holds is valid for all k ∈ H 1 (0, 1) and some constant M 1 , cf. [28, Thm. 1.4.8.1]. Take finally f ∈ V 0 and observe that by the above Nash-type inequality We have so far shown that If instead (4.1) holds, then the claim follows by duality. says that the dimension of the semigroup (T 2 (t)) t≥0 is 1. This is true regardless of the structure of the underlying graph.
Combining the stability and ultracontractivity results we have obtained, we can finally show that X 2 −X ∞ uniform exponential stability holds.
Corollary 4.5. Let the matrix B be dissipative and satisfy (3.6). Then the semigroup (T 2 (t)) t≥0 on X 2 satisfies the estimate where M is the constant that appears in (4.2) and ω is the strictly negative growth bound of (T 2 (t)) t≥0 .
Proof. Taking into account Lemma 4.3 and Theorem 4.1, the claim follows directly from [36, Lemma 6.5].
In order to discuss the well-posedness of the problem in an L p -setting, we want to identify the generators of the (T p (t)) t≥0 . Proposition 4.6. Let the matrix B be dissipative, and let it satisfy the assumptions (3.6)-(4.1). Then for all p ∈ [1, ∞] the generator A p of the semigroup (T p (t)) t≥0 is the operator whose action on the domain is formally given in (3.1). In particular, A p has compact resolvent for all p ∈ [1, ∞].
Proof. The proof goes in two steps. We first consider the case of p ∈ (2, ∞], then discuss the case p ∈ [1, 2) by duality. 1) Let p > 2. We have already remarked that X p ֒→ X q for all 1 ≤ q ≤ p ≤ ∞. Moreover, for all p > 2 the space X p is invariant under (T p (t)) t≥0 by the ultracontractivity of (T 2 (t)) t≥0 , p > 2. Thus, by [17, Prop. II.2.3] the generator of (T p (t)) t≥0 is the part of A in X p . A direct computations yields the claim.
2) Take now some p with 1 ≤ p < 2, and let q such that p −1 + q −1 = 1. By an argument similar to that of [15,Thm. 1.4.1] one has that the adjoint semigroup ((T p ) * (t)) t≥0 of (T p (t)) t≥0 on X p is actually the extrapolation semigroup ((T * ) q (t)) t≥0 of ((T * ) 2 (t)) t≥0 . Since the generator (A * ) q of ((T * ) q (t)) t≥0 also satisfies the assumption of the theorem, by 1) we deduce that Consider the operator A p whose action on D p is formally given by (3.1). We are going to show that the adjoint (A p ) * of (A p , D p ) actually agrees with (A * ) q . Then, since the generator of the pre-adjoint semigroup (T p (t)) t≥0 on X p of ((T * ) q (t)) t≥0 on X q is the pre-adjoint operator A p of A q , we conclude that (A p , D p ) generates the C 0 -semigroup (T p (t)) t≥0 on X p , and the claim follows. By definition we have that the adjoint of (A p , D p ) is the operator ((A p ) * , D * p ) given by D * p = {f ∈ X * p : ∃g ∈ X * p :< A p u, f > = < u, g > ∀u ∈ D p } = {f ∈ X q : ∃g ∈ X q :< A p u, f > = < u, g > ∀u ∈ D p }, Let us first show that D * p ⊂ D((A * ) q ). Take f ∈ D * p and observe that the identity < A p u, f > = < u, g > holds in particular for all u of the form introduced in (3.5), with u j ∈ C ∞ c (0, 1). Thus, we obtain that for all j = 1, . . . , m there exists g j ∈ L q (0, 1) such that Integrating by parts one thus obtains that (c j f ′ j ) ′ = g j (in the sense of distributions), and since g j ∈ L q (0, 1) it follows from the definition of Sobolev space of order 2 that f j ∈ W 2,q (0, 1). Thus, we conclude that f ∈ W 2,q (0, 1) m .
In order to check that the node conditions are also verified, let us perform a computation similar to that in Lemma 3.4. The condition < A p u, f > = < u, g > for u ∈ D p reads then for some g ∈ X q . Since u ∈ D p is arbitrary (and therefore so are its derivative's nodal values), we deduce that all terms on both the left and the right hand sides must vanish identically. In particular, whenever the edge e j is incident to v n the number c j (v n )φ nj u ′ j (v n ) is arbitrary (recall that c j (x) ≥ c > 0 for all x ∈ [0, 1] and j = 1, . . . , m), so that necessarily f j (v n ) = 0 for all j ∈ Γ(v n ). Now we invoke again the arbitrarity of u ∈ D p (and hence of its and its derivative's nodal values) and, considering functions u s.t. d u i = 0 for i = 1, . . . , n, we conclude that there holds Moreover, observe that the generalized Kirchhoff law for u ∈ D p ∩ (H 1 0 (0, 1)) m becomes Let us reduce (4.6) and (4.7) to pairwise relations. More precisely, pick u in such a way that only exactly two values ( Then, we obtain from (4.6)-(4.7) that for given i the vector as well as to 1 1 .

This promptly yields that
Repeating the argument m − 1 times we conclude that the nodal values f j (v i ) := d f i do not depend on j ∈ Γ(v i ). Thus, the second term on the right hand side of (4.5) can be written as or rather, taking into account the generalized Kirchhoff condition satisfied by u, as Since this expression vanishes identically and because of the arbitrarity of the nodal values of u and u ′ , we conclude that Summing up, f satisfies the Dirichlet condition at v n as well as the generalized Kirchhoff law at v 1 , . . . , v n−1 for coefficients given by B * , thus f ∈ D((A * ) q ).
Let us now check that D((A * ) q ) ⊂ D * p . Take f ∈ W 2,q (0, 1) m satisfying the continuity condition on the nodal values as well as the Dirichlet condition on v n and the generalized Kirchhoff law on v 1 , . . . , v n−1 for coefficients given by B * . Set g j = (c j f ′ j ) ′ , so that g ∈ X q . We only have to prove that for all u ∈ D p there holds < A p u, f > = < u, g >, i.e., Integrating by parts as in the proof of the converse inclusion and recalling that the nodal values of both f and u do not depend on j we see that this is the case if and only if the expression in (4.8) vanishes identically.
As a direct consequence of the ultracontractivity of (T 2 (t)) t≥0 and the Dunford-Petty Theorem, the semigroup has an integral kernel for all t > 0, cf. [4, § 7.3]. More precisely, denote by (T p (t)) t≥0 the semigroup on L p (0, m) that is similar to (T p (t)) t≥0 on X p with a similarity transformation given by the isometry U introduced in Definition 2.2. Then for all p ∈ [1, ∞] the action of (T p (t)) t≥0 is given bỹ for a suitable kernelK t ∈ L ∞ (0, m) × (0, m) . The existence of integral kernels is a typical feature of diffusion equations. Also in view of its consequences in the theory of evolution equations (see e.g. [3]), it is of great interest to estimate such kernels and compare them with the standard Gaussian one, which is associated to the heat equation on the whole space. This is usually done by the so-called Davies' trick, that amounts to prove uniform L ∞ -(quasi)contractivity estimates for a class of perturbed semigroups.
More precisely, introduce a class W of functions ϕ : Ê → Ê constructed in the following way: first, we We then stretch via the isometry U such functions over the network to functionsφ : (0, m) → Ê. We finally call W the class of all smooth, bounded continuous extensions ϕ ofφ to the whole line such that also ϕ ′ ∞ ≤ 1 and ϕ ′′ ∞ ≤ 1. Finally, for fixed ϕ ∈ W we define an operator L ρ on L 2 (Ê) by L ρ f := e −ρϕ f and perturbed semigroups (T ρ 2 (t)) t≥0 , whereT ρ 2 (t) := L ρT2 (t)L −1 ρ , ρ ∈ Ê. Observe that, by construction, In the remainder of this section we consider X p as real spaces, 1 ≤ p ≤ ∞. Then, by [7,Thm. 3.3] Gaussian estimates for (T 2 (t)) t≥0 are equivalent to ultracontractivity estimates for (T ρ 2 (t)) t≥0 , uniformly in ρ ∈ Ê and ϕ ∈ W . This can be done by applying the above presented theory to the similar semigroups (T ρ 2 (t)) t≥0 on X 2 , which by a direct computation are associated to the bilinear forms a ρ defined by for all f, g ∈ V 0 . In the following we restrict ourselves to the local case, i.e., to the case where B is a diagonal matrix. In fact, by considering the above form a ρ with coefficients c ≡ 1 and B = −1 1 1 −1 on the domain H 1 (0, 1) (i.e., for the sake of simplicity, on a graph with a single edge without Dirichlet boundary conditions), one sees that it is not possible to find ω ∈ Ê such that the shifted form a ρ +ω(1+ρ 2 ) is accretive uniformly in ρ ∈ Ê and φ ∈ W , i.e., such that for all ρ ∈ Ê, φ ∈ W , and f ∈ H 1 (0, 1). This can be checked by taking f constant, ρ = 1, φ(x) := (1+ω)x.
The uniform accretivity of the forms a ρ seems to be an essential ingredient of the method of proof explained in [7]: we thus derive in the following Gaussian estimates only in the local case of B diagonal.
Theorem 4.7. Let the matrix B be diagonal with negative entries. Then the semigroup (T 2 (t)) t≥0 has Gaussian estimates. More precisely, there exist constants b, c > 0 such thatK t satisfies uniformly in t > 0.
Proof. Under our assumptions it is already known that (T 2 (t)) t≥0 is positive and contractive with respect to the X 1 and X ∞ norm. Moreover, a direct computation shows that the shifted form a ρ + (1 + ρ 2 ) is accretive.
Observe now that V 0 is not an ideal of (H 1 (0, 1)) m , but indeed of The proof can now be concluded by mimicking [7,Thm. 4.4].
We are finally able to obtain an optimal result on the analiticity of the semigroup generated by A. We stress that we are not imposing any assumption on B.
Theorem 4.8. Consider the operator A p whose action on the domain defined in (4.4) is formally given by 3.1. Then A p generates on X p a strongly continuous, analytic semigroup (T p (t)) t≥0 of angle π 2 , p ∈ [1, ∞).
Proof. The proof goes in three steps.
1) Let us first consider the case of B = 0, p = 2. Then, it follows by Proposition 3.2 that the form a is symmetric and coercive, hence in particular the associated operator A is self-adjoint and dissipative, and the statement follows by the spectral theorem.
2) If B = 0, then for general p ∈ [1, ∞] the semigroup (T 2 (t)) t≥0 extrapolates to a semigroup (T p (t)) t≥0 on X p that is analytic of angle π 2 , by Theorem 4.7 and [36, Thm. 6.16]. 3) Finally, consider the case of general B. Then we can apply the theory developed in [18, § 2], after setting X := X p , ∂X := n−1 , and as well as and Φu := d u , u ∈ V 0 . We consider the operator A with maximal domain Y and observe that the restriction of A to ker(L) is the operator considered in 2), hence the generator of an analytic semigroup of angle π 2 . Since the boundary perturbation operator Φ : V 0 → ∂X is compact, the claim follows by [18,Thm. 2.6] (observe that in that theorem is also proved, although not explicitly stated, that the angle of analiticity remains invariant under admissible boundary perturbations).
Remark 4.9. Gaussian estimates like (4.9) are a key argument for discussing a number of different issues that go far beyond the scope of this paper. Without going into details, we recall that Theorem 4.7 implies at once, among other, the property of maximal regularity for (T p (t)) t≥0 for p ∈ (1, ∞), upper estimates for the time derivative of the kernel K t , L p -estimates for Schrödinger and wave equations, and the fact that A p has bounded H ∞ -calculus on each sector (and therefore that it has bounded imaginary powers) for p ∈ (1, ∞), cf. [36,§ 6.5 and Chapt. 7], [4, § 7.4] and references therein. Observe that, even if Gaussian estimates can only be obtained for local nodal conditions, most of the above mentioned consequences also hold in the general case by perturbation methods.
Theorem 4.10. The first order network diffusion problem introduced in Section 2 is well-posed on X p , p ∈ [1, ∞), i.e., for all initial data u 0 ∈ X p it admits a unique mild solution that continuously depends on the initial data. Such a solution is of class C ∞ in both variables x, t and its ∞-norm tends to 0 in time. If furthermore c j ∈ C ∞ [0, 1], j = 1, . . . , m, then the solution u(t, ·) is of class C ∞ with respect to the space variable.
Proof. The well-posedness and boundedness results follow from the fact that the semigroup (T 2 (t)) t≥0 is ultracontractive and extends to a family of semigroups (T 2 (p)) t≥0 that, by Proposition 4.6, actually govern the network diffusion problem. The decay of the solution is ensured by the uniform exponential stability of all semigroups.
Finally, if c j ∈ C ∞ [0, 1], j = 1, . . . , m, then one sees that D(A ∞ p ) ⊂ (C ∞ [0, 1]) m for all p ∈ (1, ∞). Since the semigroup (T p (t)) t≥0 is analytic, it maps X p into D(A ∞ p ), and the claim follows. Observe that if we replace the Dirichlet condition in v n by continuity of the values of u j (t, v n ), t ≥ 0, j ∈ Γ(v n ), plus a Kirchhoff-type condition analogous to that imposed on the other nodes, we obtain the system (4.10) x ∈ (0, 1), j = 1, . . . , m.
Here b 1h , b i1 , are arbitrary numbers, 1 ≤ i, h ≤ n − 1. Such an initial-value problem has been proved to be well-posed in [21] (in the special case of B = 0): we can compare its solution and that to the original network diffusion problem and obtain the following. For the sake of simplicity, in the following we restrict to the case of purely Kirchhoff nodal conditions. However, one can see that a similar proof also works in the general case.
Proposition 4.11. Let the coefficients b ih = 0, 1 ≤ i, h ≤ n. Then the semigroup (T 2 (t)) t≥0 governing the original network diffusion problem (as in § 2) is dominated by the semigroup (T 2 (t)) t≥0 governing (4.10) in the sense of positive semigroups.
Proof. As shown in [21], (T 2 (t)) t≥0 is a sub-Markovian semigroup that comes from a form with domain Since a Dirichlet condition in the node v n implies in particular continuity on a function in that vertex, one sees that V 0 ⊂ V . Accordingly, by [36,Cor. 2.22] it suffices to prove that V 0 is an ideal of V , i.e., that the following conditions are satisfied: • f ∈ V 0 , g ∈ V, and |g| ≤ |f | ⇒ g · signf ∈ V 0 . To check the first condition, observe that H 1 0 (0, 1) is an ideal of H 1 (0, 1), and that the continuity of the values of f ∈ H 1 (0, 1) m in the nodes is not affected by taking the absolute value of f . Let now f ∈ V 0 and g ∈ V . If |g| ≤ |f |, then in particular |g j (v n )| ≤ |f j (v n )| = 0 for all j ∈ Γ(v n ), i.e., g ∈ V 0 . Finally, since A generates a positive semigroup, V 0 is an ideal of itself and this yields that g · signf ∈ V 0 .

The heat equation on spaces of continuous functions
Also in view of applications, we are now interested to extend the previous L p -type well-posedness results to a setting where continuous functions are considered instead.
Consider the partÃ ∞ of A in the Banach spaceX ∞ := (C[0, 1]) m , whose domain is given by Lemma 5.1. The operatorÃ ∞ is sectorial onX ∞ , and it generates a compact analytic semigroup of angle π 2 . If the matrix B is a diagonal with negative entries, then such a semigroup is also positive and contractive.
Proof. 1) Let us first consider the case B = 0. Then by Theorem 4.1 and Theorem 4.8 we deduce that all the operators A p , p ∈ [1, ∞], are dissipative and sectorial of angle π 2 . In particular, for each ǫ ∈ (0, π 2 ) there exists M ǫ ≥ 1 such that the estimate with compact embedding. Therefore, we see that R(λ, A ∞ )f is a continuous function for all λ ∈ {µ ∈ : |arg µ| < π − ǫ}. It follows that the analogous of (5.1) also holds with respect to the norm ofX ∞ , henceÃ ∞ is sectorial and it generates an analytic semigroup of angle π 2 .
2) The case of general B can be treated as in the proof of Theorem 4.8, by means of the theory of boundary perturbation for sectorial operators discussed in [18].
The main motivation for considering semigroups on (C[0, 1]) m comes from applications involving semilinear equations, since we can then effectively apply the theory developed, e.g., in [27] in order to discuss well-posedness and stability. As an elementary, yet motivating example we mention the following system, related to a Hodgkin-Huxley-model describing the transmission of potential along neurons (see [38] and references therein).
Proposition 5.2. Let ψ j ∈ C 2 (Ê), j = 1, . . . , m. Consider for t > 0 the semilinear parabolic network problem given by Then for all u 0 ∈ (C[0, 1]) m the Cauchy problem associated to such a system admits a unique (global) mild solution u that depends continuously (with respect to the sup-norm) on the initial data. In fact, u satisfies the problem pointwise for t > 0.
A thorough treatment of well-posedness and stability of semilinear diffusion problems over networks goes beyond the scope of this paper. We will deal with such an issue in the forthcoming paper [13].
Even in the linear case (i.e., ψ 1 ≡ 0, j = 1, . . . , m), the problem considered in Proposition (5.2) is not well-posed in a classical sense. In fact, albeit sectorial (hence the generator of an analytic semigroup), the operatorÃ ∞ is not densely defined inX ∞ , thus the generated semigroup is not strongly continuous.
By the theorem of Stone-Weierstrass, the already defined space C 0 (G) of continuous function over the network that vanish in v n ) is the closure of D(Ã ∞ ).
Theorem 5.3. The part A ofÃ ∞ in C 0 (G) generates a compact, strongly continuous semigroup which is analytic of angle π 2 . If further B is diagonal with negative entries, then such a semigroup is contractive, real, positive, and uniformly exponentially stable.
Proof. Reasoning as in the proof of Proposition 5.2, we deduce from Theorem 4.1 that A is a resolvent positive operator on C 0 (G). Since A is also densely defined, by [6,Thm. 3.11.9] it generates a positive strongly continuous semigroup (T(t)) t≥0 .
By Lemma 5.1, we see that A is sectorial and dissipative: this yields the analyticity (with angle π 2 ) and the contractivity of (T(t)) t≥0 . Observe further that the p-independence of the spectrum of A p (by Theorem 4.1) yields the invertibility of A, hence the uniform exponential stability of (T(t)) t≥0 .
Finally, in order to show that the semigroup is compact, observe that due to its analyticity T 2 (t) maps for all t > 0. Thus, denoting by X A the Banach space obtained by endowing D(A) with the graph norm, we have Here i C 0 (G),X 2 and i X A ,C 0 (G) denote the canonical imbeddings of C 0 (G) into X 2 and of X A into C 0 (G), respectively. It follows from the theorem of Ascoli-Arzelà that the latter imbedding is compact, so that also T(t) is compact for t > 0, and the claim follows.
We can finally draw a conclusion that is similar to Theorem 4.10, and can be proved likewise.
Theorem 5.4. The network diffusion problem is well-posed on C 0 (G), i.e., for all initial data u 0 ∈ C 0 (G) it admits a unique classical solution that continuously depends on the initial data. The sup-norm of the solution tends to 0 in time.

A technical lemma
The following result seems to be of independent interest. Its proof is due to Wolfgang Arendt, whom we warmly thank. Lemma 6.1. Let A = (a ih ) be a n × n matrix with complex-valued coefficients. Then the semigroup (e tA ) t≥0 generated by A is ℓ ∞ -contractive, i.e., if and only if Proof. The proof goes in two steps. 1) Let us first assume the semigroup (e tA ) t≥0 generated by the matrix A = (a ih ) to be positive, i.e., to have real entries that are positive off-diagonal. Then it is known that (e tA ) t≥0 is ℓ ∞ -contractive if and only if A½ ≤ 0, i.e., if and only if (6.2) a ii + h =i a ih ≤ 0 for all i = 1, . . . , n.
2) Let us now consider the case of a general matrix A and define a new matrix A ♯ = (a ♯ ih ) by It is known (see [16]) that A ♯ generates the modulus semigroup of (e tA ) t≥0 , i.e., the (unique) semigroup (e tA ♯ ) t≥0 that dominates (e tA ) t≥0 in the sense of positive semigroups, and is dominated by any other semigroup also dominating (e tA ) t≥0 . Let us first assume that (6.1) holds. Since (e tA ♯ ) t≥0 is positive, by part 1) it is also ℓ ∞ -contractive. Now, since (e tA ♯ ) t≥0 dominates (e tA ) t≥0 , it follows that (e tA ) t≥0 is ℓ ∞ -contractive as well.