Dynamic and generalized Wentzell node conditions for network equations

Mugnolo, Delio; Romanelli, Silvia

doi:10.1002/mma.805

Cited by 43 publications

(43 citation statements)

References 35 publications

(24 reference statements)

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“…In the case of finite V 0 [clearly, a special case of (1)], well-posedness has already been observed in [53, § 19] and also in [37].…”

Section: The Laplacian On a Networkmentioning

confidence: 79%

Asymptotics of semigroups generated by operator matrices

Mugnolo

2014

Arab. J. Math.

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We survey some known results about operator semigroup generated by operator matrices with diagonal or coupled domain. These abstract results are applied to the characterization of well-/ill-posedness for a class of evolution equations with dynamic boundary conditions on domains or metric graphs. In particular, our results on the heat equation with general Wentzell-type boundary conditions complement those previously obtained by, among others, Bandle-von Below-Reichel and Vázquez-Vitillaro.

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“…In the case of finite V 0 [clearly, a special case of (1)], well-posedness has already been observed in [53, § 19] and also in [37].…”

Section: The Laplacian On a Networkmentioning

confidence: 79%

Asymptotics of semigroups generated by operator matrices

Mugnolo

2014

Arab. J. Math.

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“…The diffusion of electric potential along every edge of the graph thus follows the equation (1.2). We avoid to introduce more general network elliptic operators: in fact, under uniform ellipticity assumptions this case is known to present no serious mathematical challenges over the basic case of a plain network Laplacian: compare [16].…”

Section: Abstract Setting For the Neural Network Modelmentioning

confidence: 99%

Existence and stability of square-mean almost periodic solutions to a spatially extended neural network with impulsive noise

Bonaccorsi

Ziglio

2014

Random Operators and Stochastic Equations

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Abstract. Our work is concerned with a neural network with n nodes, where the activity of the k-th cell depends on external, stochastic inputs as well as the coupling generated by the activity of the adjacent cells, transmitted through a diffusion process in the network. This paper aims to throw some light on time-varying, stochastically perturbed, neuronal networks. We show that when the coefficients oscillate around a reference value, with ascillations that are almost periodic and suitably small in percentage, then there exists a unique solution for the system, that is almost periodic and uniformly bounded in the mean square norm for all times.

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“…Since we were not primarily motivated by specific applications we refer to [17,18,23] for some of these motivations. An evident motivation is the description of (electric) currents in networks.…”

Section: Introductionmentioning

confidence: 99%

Dirichlet forms for singular one-dimensional operators and on graphs

Kant

Klauß²,

Voigt

et al. 2009

J. Evol. Equ.

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We treat the time evolution of states on a finite directed graph, with singular diffusion on the edges of the graph and glueing conditions at the vertices. The operator driving the evolution is obtained by the method of quadratic forms on a suitable Hilbert space. Using the Beurling-Deny criteria we describe glueing conditions leading to positive and to submarkovian semigroups, respectively. IntroductionThe intentions of this paper are twofold. The first aim is to present a treatment of one-dimensional "singular" diffusion in the framework of Dirichlet forms. The second is to present suitable boundary or glueing conditions on graphs (quantum graphs) leading to positive or submarkovian C 0 -semigroups.Concerning the first topic we assume that µ is a finite Borel measure on a bounded interval [a, b]. We assume that particles move in [a, b] according to "Brownian motion" but are only allowed to be located in the support of µ, and in fact are slowed down or accelerated by the "speed measure" µ. (Incidentally, the support of µ is allowed to have gaps, what sometimes is referred to by "gap diffusion".) More generally, instead of starting with Brownian motion, one also can include a drift in the diffusion. This leads to including a scale function. The treatment of the corresponding process has a longer history (cf. [1,3-5,12-14,19,20,22]), but there appears to be no treatment of the arising evolution in the context of Dirichlet forms.Concerning the second topic, we assume that finitely many intervals, with diffusion as described above, are arranged in a graph, and we treat the question how boundary conditions (glueing conditions) at the vertices can be posed in a way to describe diffusing particles. These topics have also been treated in a recent paper by Kostrykin et al. [16]. Since we pose the glueing conditions in a different form (following [17]) it does not seem evident to us to establish the connection between the conditions given in [16] and our conditions.There are many motivations from applications for this work. Since we were not primarily motivated by specific applications we refer to [17,18,23] for some of these Mathematics Subject Classification (2000): 47D06, 60J60, 47E05, 35Q99, 05C99

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Dynamic and generalized Wentzell node conditions for network equations

Cited by 43 publications

References 35 publications

Asymptotics of semigroups generated by operator matrices

Asymptotics of semigroups generated by operator matrices

Existence and stability of square-mean almost periodic solutions to a spatially extended neural network with impulsive noise

Dirichlet forms for singular one-dimensional operators and on graphs

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