# American Institute of Mathematical Sciences

ISSN:
1556-1801

eISSN:
1556-181X

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## Networks & Heterogeneous Media

December 2016 , Volume 11 , Issue 4

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2016, 11(4): 545-562 doi: 10.3934/nhm.2016009 +[Abstract](2666) +[PDF](408.1KB)
Abstract:
The goal of this paper is to analyze, through homogenization techniques, the effective thermal transfer in a periodic composite material formed by two constituents, separated by an imperfect interface where both the temperature and the flux exhibit jumps. Following the hypotheses on the flux jump, two different homogenized problems are obtained. These problems capture in various ways the influence of the jumps: in the homogenized coefficients, in the right-hand side of the homogenized problem, and in the correctors.
2016, 11(4): 563-601 doi: 10.3934/nhm.2016010 +[Abstract](2629) +[PDF](649.7KB)
Abstract:
In this paper, we address the stability of transport systems and wave propagation on networks with time-varying parameters. We do so by reformulating these systems as non-autonomous difference equations and by providing a suitable representation of their solutions in terms of their initial conditions and some time-dependent matrix coefficients. This enables us to characterize the asymptotic behavior of solutions in terms of such coefficients. In the case of difference equations with arbitrary switching, we obtain a delay-independent generalization of the well-known criterion for autonomous systems due to Hale and Silkowski. As a consequence, we show that exponential stability of transport systems and wave propagation on networks is robust with respect to variations of the lengths of the edges of the network preserving their rational dependence structure. This leads to our main result: the wave equation on a network with arbitrarily switching damping at external vertices is exponentially stable if and only if the network is a tree and the damping is bounded away from zero at all external vertices but at most one.
2016, 11(4): 603-625 doi: 10.3934/nhm.2016011 +[Abstract](2783) +[PDF](536.7KB)
Abstract:
In this paper we present a suitable mathematical model to describe the behaviour of a hybrid electrolyte-oxide-semiconductor (EOS) device, that could be considered for application to neuro-prothesis and bio-devices. In particular, we discuss the existence and uniqueness of solutions also including the effects of the size exclusion in narrow structures such as ionic channels or nanopores. The result is proved using a fixed point argument on the whole domain.
Our results provide information about the charge distribution and the potential behaviour on the device domain, and can represent a suitable framework for the development of stable numerical tools for innovative nanodevice modelling.
2016, 11(4): 627-653 doi: 10.3934/nhm.2016012 +[Abstract](2546) +[PDF](506.3KB)
Abstract:
This paper concerns the homogenization of nonlinear dissipative hyperbolic problems \begin{gather*} \partial _{tt}u^{\varepsilon }\left( x,t\right) -\nabla \cdot \left( a\left( \frac{x}{\varepsilon ^{q_{1}}},\ldots ,\frac{x}{\varepsilon ^{q_{n}}},\frac{t }{\varepsilon ^{r_{1}}},\ldots ,\frac{t}{\varepsilon ^{r_{m}}}\right) \nabla u^{\varepsilon }\left( x,t\right) \right) \\ +g\left( \frac{x}{\varepsilon ^{q_{1}}},\ldots ,\frac{x}{\varepsilon ^{q_{n}} },\frac{t}{\varepsilon ^{r_{1}}},\ldots ,\frac{t}{\varepsilon ^{r_{m}}} ,u^{\varepsilon }\left( x,t\right) ,\nabla u^{\varepsilon }\left( x,t\right) \right) =f(x,t) \end{gather*} where both the elliptic coefficient $a$ and the dissipative term $g$ are periodic in the $n+m$ first arguments where $n$ and $m$ may attain any non-negative integer value. The homogenization procedure is performed within the framework of evolution multiscale convergence which is a generalization of two-scale convergence to include several spatial and temporal scales. In order to derive the local problems, one for each spatial scale, the crucial concept of very weak evolution multiscale convergence is utilized since it allows less benign sequences to attain a limit. It turns out that the local problems do not involve the dissipative term $g$ even though the homogenized problem does and, due to the nonlinearity property, an important part of the work is to determine the effective dissipative term. A brief illustration of how to use the main homogenization result is provided by applying it to an example problem exhibiting six spatial and eight temporal scales in such a way that $a$ and $g$ have disparate oscillation patterns.
2016, 11(4): 655-692 doi: 10.3934/nhm.2016013 +[Abstract](2920) +[PDF](658.5KB)
Abstract:
The large time decay rates of a transmission problem coupling heat and wave equations on a planar network is discussed.
When all edges evolve according to the heat equation, the uniform exponential decay holds. By the contrary, we show the lack of uniform stability, based on a Geometric Optics high frequency asymptotic expansion, whenever the network involves at least one wave equation.
The (slow) decay rate of this system is further discussed for star-shaped networks. When only one wave equation is present in the network, by the frequency domain approach together with multipliers, we derive a sharp polynomial decay rate. When the network involves more than one wave equation, a weakened observability estimate is obtained, based on which, polynomial and logarithmic decay rates are deduced for smooth initial conditions under certain irrationality conditions on the lengths of the strings entering in the network. These decay rates are intrinsically determined by the wave equations entering in the system and are independent on the heat equations.
2016, 11(4): 693-719 doi: 10.3934/nhm.2016014 +[Abstract](2689) +[PDF](498.7KB)
Abstract:
We consider a SIS system with nonlocal diffusion which is the continuous version of a discrete model for the propagation of epidemics on a metapopulation network. Under the assumption of limited transmission, we prove the global existence of a unique solution for any diffusion coefficients. We investigate the existence of an endemic equilibrium and prove its linear stability, which corresponds to the loss of stability of the disease-free equilibrium. In the case of equal diffusion coefficients, we reduce the system to a Fisher-type equation with nonlocal diffusion, which allows us to study the large time behaviour of the solutions. We show large time convergence to either the disease-free or the endemic equilibrium.

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