20 research outputs found
Long-time behavior of a nonlocal Cahn-Hilliard equation with reaction
In this paper we study the long-time behavior of a nonlocal Cahn-Hilliard
system with singular potential, degenerate mobility, and a reaction term. In
particular, we prove the existence of a global attractor with finite fractal
dimension, the existence of an exponential attractor, and convergence to
equilibria for two physically relevant classes of reaction terms
Canards in a bottleneck
In this paper we investigate the stationary profiles of a nonlinear
Fokker-Planck equation with small diffusion and nonlinear in- and outflow
boundary conditions. We consider corridors with a bottleneck whose width has a
global nondegenerate minimum in the interior. In the small diffusion limit the
profiles are obtained constructively by using methods from geometric singular
perturbation theory (GSPT). We identify three main types of profiles
corresponding to: (i) high density in the domain and a boundary layer at the
entrance, (ii) low density in the domain and a boundary layer at the exit, and
(iii) transitions from high density to low density inside the bottleneck with
boundary layers at the entrance and exit. Interestingly, solutions of the last
type involve canard solutions generated at the narrowest point of the
bottleneck. We obtain a detailed bifurcation diagram of these solutions in
terms of the in- and outflow rates. The analytic results based on GSPT are
further corroborated by computational experiments investigating corridors with
bottlenecks of variable width.Comment: arXiv admin note: text overlap with arXiv:2010.1442
Singular perturbation analysis of a regularized MEMS model
Micro-Electro Mechanical Systems (MEMS) are defined as very small structures
that combine electrical and mechanical components on a common substrate. Here,
the electrostatic-elastic case is considered, where an elastic membrane is
allowed to deflect above a ground plate under the action of an electric
potential, whose strength is proportional to a parameter . Such
devices are commonly described by a parabolic partial differential equation
that contains a singular nonlinear source term. The singularity in that term
corresponds to the so-called "touchdown" phenomenon, where the membrane
establishes contact with the ground plate. Touchdown is known to imply the
non-existence of steady state solutions and blow-up of solutions in finite
time. We study a recently proposed extension of that canonical model, where
such singularities are avoided due to the introduction of a regularizing term
involving a small "regularization" parameter . Methods from
dynamical systems and geometric singular perturbation theory, in particular the
desingularization technique known as "blow-up", allow for a precise description
of steady-state solutions of the regularized model, as well as for a detailed
resolution of the resulting bifurcation diagram. The interplay between the two
main model parameters and is emphasized; in particular,
the focus is on the singular limit as both parameters tend to zero
Travelling waves due to negative plant-soil feedbacks in a model including tree life-stages
The emergence and maintenance of tree species diversity in tropical forests is commonly attributed to the Janzen-Connell (JC) hypothesis, which states that growth of seedlings is suppressed in the proximity of conspecific adult trees. As a result, a JC distribution due to a density-dependent negative feedback emerges in the form of a (transient) pattern where conspecific seedling density is highest at intermediate distances away from parent trees. Several studies suggest that the required density-dependent feedbacks behind this pattern could result from interactions between trees and soil-borne pathogens. However, negative plant-soil feedback may involve additional mechanisms, including the accumulation of autotoxic compounds generated through tree litter decomposition. An essential task therefore consists in constructing mathematical models incorporating both effects showing the ability to support the emergence of JC distributions. In this work, we develop and analyse a novel reaction-diffusion-ODE model, describing the interactions within tropical tree species across different life stages (seeds, seedlings, and adults) as driven by negative plant-soil feedback. In particular, we show that under strong negative plant-soil feedback travelling wave solutions exist, creating transient distributions of adult trees and seedlings that are in agreement with the Janzen-Connell hypothesis. Moreover, we show that these travelling wave solutions are pulled fronts and a robust feature as they occur over a broad parameter range. Finally, we calculate their linear spreading speed and show its (in)dependence on relevant nondimensional parameters
Modelling how negative plant-soil feedbacks across life stages affect the spatial patterning of trees
In this work, we theoretically explore how litter decomposition processes and soil-borne pathogens contribute to negative plant-soil feedbacks, in particular in transient and stable spatial organisation of tropical forest trees and seedlings known as Janzen-Connell distributions. By considering soil-borne pathogens and autotoxicity both separately and in combination in a phenomenological model, we can study how both factors may affect transient dynamics and emerging Janzen-Connell distributions. We also identify parameter regimes associated with different long-term behaviours. Moreover, we compare how the strength of negative plant-soil feedbacks was mediated by tree germination and growth strategies, using a combination of analytical approaches and numerical simulations. Our interdisciplinary investigation, motivated by an ecological question, allows us to construct important links between local feedbacks, spatial self-organisation, and community assembly. Our model analyses contribute to understanding the drivers of biodiversity in tropical ecosystems, by disentangling the abilities of two potential mechanisms to generate Janzen-Connell distributions. Furthermore, our theoretical results may help guiding future field data analyses by identifying spatial signatures in adult tree and seedling distribution data that may reflect the presence of particular plant-soil feedback mechanisms
Geometrische Analysis von Mehrskalenlösungen in regularisierten Modellen fĂŒr Mikrostrukturen und Touchdown PhĂ€nomenen in MEMS
Abweichender Titel nach Ăbersetzung der Verfasserin/des VerfassersDie Analysis qualitativer Eigenschaften von nichtlinearen partiellen Differentialgleichungen (PDEs) durch Methoden aus der Theorie dynamischer Systeme ist ein aktives Forschungsgebiet. Ein wichtiges Thema dabei ist die Analyse der Existenz, StabilitĂ€t und Verzweigung von speziellen Lösungen, die wesentliche Merkmale der zu untersuchenden PDE enthalten. Insbesondere fĂŒr PDEs mit singulĂ€ren Störungen oder SingularitĂ€ten hat sich dabei eine Kombination aus Methoden aus der Theorie dynamischer Systeme, Methoden der singulĂ€ren Störungstheorie und numerischen Berechnungen als sehr effektiv erwiesen. Dieses Projekt befasst sich mit zwei Problemen dieser Art, die neuartige Multiskalenmerkmale aufweisen. Im ersten Problem untersuchen wir die Euler-Lagrange-Gleichung der Regularisierung eines nichtkonvexen Variationsproblems, das als einfaches mathematisches Modell fĂŒr Mikrostrukturen in FormgedĂ€chtnislegierungen auftritt. FĂŒr dieses singulĂ€r gestörte Hamiltonsche System beweisen wir die Existenz einer Klasse von periodischen Lösungen und untersuchen ihre AbhĂ€ngigkeit von den wesentlichen Parametern durch asymptotische Methoden und numerische Fortsetzung. Das Ziel ist ein besseres VerstĂ€ndnis der Struktur von minimierenden Lösungen und ihres ungewöhnlichen Skalierungsverhaltens. Mittels numerischer Pfadverfolgung werden zusĂ€tzlich neue Typen von Lösungen gefunden. Das zweite Problem betrifft die Asymptotik und Verzweigung von stationĂ€ren Lösungen eines Modells fĂŒr mikroelektromechanische Systeme (MEMS). Dieses Modell wurde kĂŒrzlich als Regularisierung eines einfacheren Modells vorgeschlagen, von dem bekannt ist, dass es in endlicher Zeit SingularitĂ€ten entwickelt. FĂŒr dieses Problem wird das numerisch berechnete Verzweigungsdiagramm erklĂ€rt, indem die Interaktion des Regularisierungsterms mit der fĂŒr das Touchdown-PhĂ€nomen verantwortlichen SingularitĂ€t im Detail untersucht wird. Dabei wird die Blow-up-Methode verwendet, um die Dynamik in der NĂ€he der SingularitĂ€t zu analysieren und rigorose Ergebnisse zu erhalten, welche die bereits existierende formale Asymptotik und Numerik beweisen und ergĂ€nzen. Ein zentraler aspekt dabei ist die Untersuchung einer speziellen Sattel-Knoten Verzweigung, deren numerische Untersuchung aufgrund ihres singulĂ€ren Charakters sehr schwierig ist. Diese neuartige Erweiterung der Blow-up-Methode zur Analyse von Randwertproblemen und singulĂ€ren Grenzenwerten in Verzweigungsproblemen hat das Potential auch in anderen ZusammenhĂ€ngen von Nutzen zu sein.11
Single-spike solutions to the 1D shadow Gierer-Meinhardt problem
A fundamental example of reaction-diffusion system exhibiting Turing type
pattern formation is the Gierer-Meinhardt system, which reduces to the shadow
Gierer-Meinhardt problem in a suitable singular limit. Thanks to its
applicability in a large range of biological applications, this singularly
perturbed problem has been widely studied in the last few decades via rigorous,
asymptotic, and numerical methods. However, standard matched asymptotics
methods do not apply (Ni 1998, Wei 1998), and therefore analytical expressions
for single spike solutions are generally lacking.
By introducing an ansatz based on generalized hyperbolic functions, we
determine exact radially symmetric solutions to the one-dimensional shadow
Gierer-Meinhardt problem for any , representing both inner and
boundary spike solutions depending on the location of the peak. Our approach
not only confirms numerical results existing in literature, but also provides
guidance for tackling extensions of the shadow Gierer-Meinhardt problem based
on different boundary conditions (e.g. mixed) and/or -dimensional domains
On wave propagation in nanobeams
Wave propagation in Rayleigh nanobeams resting on nonlocal media is investigated in this paper. Small-scale structure-foundation problems are formulated according to a novel consistent nonlocal approach extending the special elastostatic analysis in Barretta et al. (2022). Nonlocal effects of the nanostructure are modelled according to a stress-driven integral law. External elasticity of the nano-foundation is instead described by a displacement-driven spatial convolution. The developed methodology leads to well-posed continuum problems, thus circumventing issues and applicative difficulties of the EringenâWieghardt nonlocal approach. Wave propagation in Rayleigh nanobeams interacting with nano-foundations is then analysed and dispersive features are analytically detected exploiting the novel consistent strategy. Closed form expressions of size-dependent dispersion relations are established and connection with outcomes available in literature is contributed. A general and well-posed methodology is thus provided to address wave propagation nanomechanical problems. Parametric studies are finally accomplished and discussed to show effects of length scale parameters on wave dispersion characteristics of small-scale systems of current interest in Nano-Engineering