Asymptotic Behavior for a Nonlocal Diffusion Equation in Domains with Holes

The paper deals with the asymptotic behavior of solutions to a non-local diffusion equation, $u_t=J*u-u:=Lu$, in an exterior domain, $\Omega$, which excludes one or several holes, and with zero Dirichlet data on $\mathbb{R}^N\setminus\Omega$. When the space dimension is three or more this behavior is given by a multiple of the fundamental solution of the heat equation away from the holes. On the other hand, if the solution is scaled according to its decay factor, close to the holes it behaves like a function that is $L$-harmonic, $Lu=0$, in the exterior domain and vanishes in its complement. The height of such a function at infinity is determined through a matching procedure with the multiple of the fundamental solution of the heat equation representing the outer behavior. The inner and the outer behavior can be presented in a unified way through a suitable global approximation

Boundary fluxes for non-local diffusion

We study a nonlocal diffusion operator in a bounded smooth domain prescribing the flux through the boundary. This problem may be seen as a generalization of the usual Neumann problem for the heat equation. First, we prove existence, uniqueness and a comparison principle. Next, we study the behavior of solutions for some prescribed boundary data including blowing up ones. Finally, we look at a nonlinear flux boundary condition

Ligninolytic enzyme activities and colonization by Anthracophyllum discolor on lignocellulosic supports

FONDECYT Nº109067

A nonlocal inhomogeneous dispersal process

AbstractThis article in devoted to the study of the nonlocal dispersal equationut(x,t)=∫RJ(x−yg(y))u(y,t)g(y)dy−u(x,t)in R×[0,∞), and its stationary counterpart. We prove global existence for the initial value problem, and under suitable hypothesis on g and J, we prove that positive bounded stationary solutions exist. We also analyze the asymptotic behavior of the finite mass solutions as t→∞, showing that they converge locally to zero

Selection of white-rot fungi to formulate complex and coated pellets for Reactive Orange 165 decolourization

Six strains of white-rot fungi isolated from southern Chile were evaluated for their ergosterol/biomass correlation and ligninolytic potential in solid medium to formulate pellets for Reactive Orange 165 (RO165) decolourization. The fungus Anthracophyllum discolor was selected to formulate complex pellets (fungal mycelium, sawdust, and activated carbon), coated pellets (complex pellet + alginate) and simple pellets (fungal mycelium). The activity of ligninolytic enzymes (laccase, manganese peroxidase, manganese-independent peroxidase, and lignin peroxidase) was evaluated in both the complex and coated pellets in modified Kirk medium, and the morphology of the pellets was studied using scanning electron microscopy (SEM). Complex pellets of A. discolor showed a higher enzymatic production mainly MnP (38 U L-1 at day 15) compared to coated and simple pellets. Examinations using SEM showed that both pellets produced a black core that was entrapped by a layer of fungal mycelium. Decolourization of RO165 was demonstrated with all the pellets formulated. However, the highest and fastest decolourization was obtained with complex pellets (100% at day 8). Therefore, complex pellets of A. discolor can be used for the biological treatment of wastewater contaminated with RO165.This research was supported by a FONDECYT grant 1090678 and a Doctoral thesis fellowship CONICYT 24100149

2-Vector Spaces and Groupoids

This paper describes a relationship between essentially finite groupoids and 2-vector spaces. In particular, we show to construct 2-vector spaces of Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation--a weak functor--from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail.Comment: 44 pages, 5 figures - v2 adds new theorem, significant changes to proofs, new sectio

Native forest replacement by exotic plantations in southern Chile (1985–2011) and partial compensation by natural regeneration

Although several studies have reported rates of deforestation and spatial patterns of native forest fragmentation, few have focused on the role of natural forest regeneration and exotic tree plantations on landscape dynamics. The objective of this study was to analyze the dynamics of land cover change in order to test the hypothesis that exotic tree plantations have caused a major transformation of temperate forest cover in southern Chile during the last three decades. We used three Landsat satellite images taken in 1985 (TM), 1999 (ETM+), and 2011 (TM) to quantify land cover change, together with a set of landscape indicators to describe the spatial configuration of land cover. Our results showed that the major changes were dynamic conversion among forest, exotic tree plantation and shrubland. During the study period, the area covered by exotic tree plantations increased by 168% (20,896–56,010 ha), at an annual rate of 3.8%, mostly at the expense of native forest and shrubland. There was a total gross loss of native forest of 30% (54,304 ha), but a net loss of initial cover of only 5.1% (9130 ha), at an annual net deforestation rate of 0.2%. The difference between gross and net loss of native forest was mostly the result of conversion of shrubland and agricultural and pasture land to secondary forest following natural regeneration. Over the course of the study period, exotic tree plantations showed a constant increase in patch density, total edge length, nearest-neighbor distance, and largest patch index; maximum mean patch size occurred in the middle of the study period. Native forest exhibited an increase and then a decrease in patch density and total edge length, whereas mean patch size and largest patch index were lowest in the middle of the period. Overall, the observed trends indicate expansion of exotic tree plantations and increase in native forest loss and fragmentation, particularly between 1985 and 1999. Forest loss included both old-growth and secondary forests, while native forest established after secondary succession differed in diversity, structure, and functionality from old-growth and old growth/secondary forests. Since different successional stages influence the provision of ecosystem services, the changes observed in our study are likely to have consequences for humans that extend beyond immediate changes in land use patterns