pinecone/core-2020-05-10-deduplication huggingface dataset:
Long-term behavior of reaction-diffusion equations with nonlocal
  boundary conditions on rough domains

We investigate the long term behavior in terms of finite dimensional global
and exponential attractors, as time goes to infinity, of solutions to a
semilinear reaction-diffusion equation on non-smooth domains subject to
nonlocal Robin boundary conditions, characterized by the presence of fractional
diffusion on the boundary. Our results are of general character and apply to a
large class of irregular domains, including domains whose boundary is Holder
continuous and domains which have fractal-like geometry. In addition to
recovering most of the existing results on existence, regularity, uniqueness,
stability, attractor existence, and dimension, for the well-known
reaction-diffusion equation in smooth domains, the framework we develop also
makes possible a number of new results for all diffusion models in other
non-smooth settings.Comment: 39 pages, 3 figures. arXiv admin note: text overlap with
  arXiv:0901.4412 by other author
New results on averaging theory and applications

Agraïments: The first author is supported by CNPq 248501/2013-5. CAPES grant 88881.030454 /2013-01 from the Program CSF-PVEThe usual averaging theory reduces the computation of some periodic solutions of a system of ordinary differential equations, to find the simple zeros of an associated averaged function. When one of these zeros is not simple, i.e. the Jacobian of the averaged function in it is zero, the classical averaging theory does not provide information about the periodic solution associated to a non simple zero. Here we provide sufficient conditions in order that the averaging theory can be applied also to non simple zeros for studying their associated periodic solutions. Additionally we do two applications of this new result for studying the zero--Hopf bifurcation in the Lorenz system and in the Fitzhugh--Nagumo system
Exact solutions for logistic reaction-diffusion in biology

Reaction-diffusion equations with a nonlinear source have been widely used to
model various systems, with particular application to biology. Here, we provide
a solution technique for these types of equations in $N$-dimensions. The
nonclassical symmetry method leads to a single relationship between the
nonlinear diffusion coefficient and the nonlinear reaction term; the subsequent
solutions for the Kirchhoff variable are exponential in time (either growth or
decay) and satisfy the linear Helmholtz equation in space. Example solutions
are given in two dimensions for particular parameter sets for both quadratic
and cubic reaction terms.Comment: 18 pages, 6 figure
Evolutionary dynamics of phenotype-structured populations : from individual-level mechanisms to population-level consequences

This research was supported in part by the Australian Research Council (DP140100339) and by the French National Research Agency through the ANR blanche project Kibord [ANR-13-BS01-0004] and the “ANR JC” project Modevol [ANR-13-JS01-0009]. TL was also supported in part by the Hadamard Mathematics Labex, backed by the Fondation Mathématique Jacques Hadamard, through a grant overseen by the French National Research Agency [ANR-11-LABX-0056-LMH]. LD was also supported in part by Université Sorbonne Paris Cité “Investissements d’Avenir”[ANR-11-IDEX-0005].Epigenetic mechanisms are increasingly recognised as integral to the adaptation of species that face environmental changes. In particular, empirical work has provided important insights into the contribution of epigenetic mechanisms to the persistence of clonal species, from which a number of verbal explanations have emerged that are suited to logical testing by proof-of-concept mathematical models. Here, we present a stochastic agent-based model and a related deterministic integrodifferential equation model for the evolution of a phenotype-structured population composed of asexually-reproducing and competing organisms which are exposed to novel environmental conditions. This setting has relevance to the study of biological systems where colonising asexual populations must survive and rapidly adapt to hostile environments, like pathogenesis, invasion and tumour metastasis. We explore how evolution might proceed when epigenetic variation in gene expression can change the reproductive capacity of individuals within the population in the new environment. Simulations and analyses of our models clarify the conditions under which certain evolutionary paths are possible, and illustrate that whilst epigenetic mechanisms may facilitate adaptation in asexual species faced with environmental change, they can also lead to a type of “epigenetic load” and contribute to extinction. Moreover, our results offer a formal basis for the claim that constant environments favour individuals with low rates of stochastic phenotypic variation. Finally, our model provides a “proof of concept” of the verbal hypothesis that phenotypic stability is a key driver in rescuing the adaptive potential of an asexual lineage, and supports the notion that intense selection pressure can, to an extent, offset the deleterious effects of high phenotypic instability and biased epimutations, and steer an asexual population back from the brink of an evolutionary dead end.PostprintPeer reviewe
Integral potential method for a transmission problem with Lipschitz
  interface in ${\mathbb R}^3$ for the Stokes and Darcy-Forchheimer-Brinkman
  PDE systems

The purpose of this paper is to obtain existence and uniqueness results in
weighted Sobolev spaces for transmission problems for the non-linear
Darcy-Forchheimer-Brinkman system and the linear Stokes system in two
complementary Lipschitz domains in ${\mathbb R}^3$, one of them is a bounded
Lipschitz domain $\Omega $ with connected boundary, and another one is the
exterior Lipschitz domain ${\mathbb R}^3\setminus \overline{\Omega }$. We
exploit a layer potential method for the Stokes and Brinkman systems combined
with a fixed point theorem in order to show the desired existence and
uniqueness results, whenever the given data are suitably small in some weighted
Sobolev spaces and boundary Sobolev spaces.Comment: 29 page
Multiple Liquid Bridges with Non-Smooth Interfaces

We consider a coexistence of two axisymmetric liquid bridges LB_i and LB_m of
two immiscible liquids i and m which are immersed in a third liquid (or gas) e
and trapped between two smooth solid bodies with axisymmetric surfaces S_1,S_2
and free contact lines. Evolution of liquid bridges allows two different
configurations of LB_i and LB_m with multiple (five or three) interfaces of
non-smooth shape. We formulate a variational problem with volume constraints
and present its governing equations supplemented by boundary conditions. We
find a universal relationship between curvature of the interfaces and discuss
the Young relation at the singular curve where all liquids meet together.Comment: 14 pages, 4 Figure
Neumann Boundary Controllability of the Gear--Grimshaw System With
  Critical Size Restrictions on the Spacial Domain

In this paper we study the boundary controllability of the Gear-Grimshaw
system posed on a finite domain $(0,L)$, with Neumann boundary conditions:
\begin{equation} \label{abs} \begin{cases} u_t + uu_x+u_{xxx} + a v_{xxx} +
a_1vv_x+a_2 (uv)_x =0, & \text{in} \,\, (0,L)\times (0,T), c v_t +rv_x
+vv_x+abu_{xxx} +v_{xxx}+a_2buu_x+a_1b(uv)_x =0, & \text{in} \,\, (0,L)\times
(0,T), u_{xx}(0,t)=h_0(t),\,\,u_x(L,t)=h_1(t),\,\,u_{xx}(L,t)=h_2(t), &
\text{in} \,\, (0,T),
v_{xx}(0,t)=g_0(t),\,\,v_x(L,t)=g_1(t),\,\,v_{xx}(L,t)=g_2(t), & \text{in} \,\,
(0,T), u(x,0)= u^0(x), \quad v(x,0)= v^0(x), & \text{in} \,\, (0,L).\nonumber
\end{cases} \end{equation} We first prove that the corresponding linearized
system around the origin is exactly controllable in $(L^2(0,L))^2$ when
$h_2(t)=g_2(t)=0$. In this case, the exact controllability property is derived
for any $L>0$ with control functions $h_0, g_0\in H^{-\frac{1}{3}}(0,T)$ and
$h_1, g_1\in L^2(0,T)$. If we change the position of the controls and consider
$h_0(t)=h_2(t)=0$ (resp. $g_0(t)=g_2(t)=0)$ we obtain the result with control
functions $g_0, g_2\in H^{-\frac{1}{3}}(0,T)$ and $h_1, g_1\in L^2(0,T)$ if and
only if the length $L$ of the spatial domain $(0,L)$ belongs to a countable
set. In all cases the regularity of the controls are sharp in time. If only one
control act in the boundary condition, $h_0(t)=g_0(t)=h_2(t)=g_2(t)=0$ and
$g_1(t)=0$ (resp. $h_1(t)=0$), the linearized system is proved to be exactly
controllable for small values of the length $L$ and large time of control $T$.
Finally, the nonlinear system is shown to be locally exactly controllable via
the contraction mapping principle, if the associated linearized systems are
exactly controllable.Comment: 30 Page
Analytic solutions for the approximated 1-D Kantorovich mass transfer
  problems

This paper mainly investigates the approximation of a global maximizer of the
1-D Monge-Kantorovich mass transfer problem through the approach of nonlinear
differential equations with Dirichlet boundary. Using an approximation
mechanism, the primal maximization problem can be transformed into a sequence
of minimization problems. By applying the canonical duality theory, one is able
to derive a sequence of analytic solutions for the minimization problems. In
the final analysis, the convergence of the sequence to a global maximizer of
the primal Monge-Kantorovich problem will be demonstrated.Comment: 10 pages, 0 figure. arXiv admin note: text overlap with
  arXiv:1607.06554, arXiv:1607.0655
Continuum Eigenmodes in Some Linear Stellar Models

We apply parallel approaches in the study of continuous spectra to adiabatic
stellar models. We seek continuum eigenmodes for the LAWE formulated as both
finite difference and linear differential equations. In particular, we apply
methods of Jacobi matrices and methods of subordinancy theory in these
respective formulations. We find certain pressure-density conditions which
admit positive-measured sets of continuous oscillation spectra under plausible
conditions on density and pressure. We arrive at results of unbounded
oscillations and computational or, perhaps, dynamic instability
Colombeau algebra as a mathematical tool for investigating step load and
  step deformation of systems of nonlinear springs and dashpots

The response of mechanical systems composed of springs and dashpots to a step
input is of eminent interest in the applications. If the system is formed by
linear elements, then its response is governed by a system of linear ordinary
differential equations, and the mathematical method of choice for the analysis
of the response of such systems is the classical theory of distributions.
However, if the system contains nonlinear elements, then the classical theory
of distributions is of no use, since it is strictly limited to the linear
setting. Consequently, a question arises whether it is even possible or
reasonable to study the response of nonlinear systems to step inputs. The
answer is positive. A mathematical theory that can handle the challenge is the
so-called Colombeau algebra. Building on the abstract result by (Pr\r{u}\v{s}a
& Rajagopal 2016, Int. J. Non-Linear Mech) we show how to use the theory in the
analysis of response of a simple nonlinear mass--spring--dashpot system
A cross-diffusion system derived from a Fokker-Planck equation with
  partial averaging

A cross-diffusion system for two compoments with a Laplacian structure is
analyzed on the multi-dimensional torus. This system, which was recently
suggested by P.-L. Lions, is formally derived from a Fokker-Planck equation for
the probability density associated to a multi-dimensional It\={o} process,
assuming that the diffusion coefficients depend on partial averages of the
probability density with exponential weights. A main feature is that the
diffusion matrix of the limiting cross-diffusion system is generally neither
symmetric nor positive definite, but its structure allows for the use of
entropy methods. The global-in-time existence of positive weak solutions is
proved and, under a simplifying assumption, the large-time asymptotics is
investigated
Determination of Gap Solution and Critical Temperature in Doped Graphene
  Superconductivity

It is shown that the gap solution and critical transition temperature are
significantly enhanced by doping in a recently developed BCS formalism for
graphene superconductivity in such a way that positive gap and transition
temperature both occur in arbitrary pairing coupling as far as doping is
present. The analytic construction of the BCS gap and transition temperature
offers highly effective globally convergent iterative methods for the
computation of these quantities. A series of numerical examples are presented
as illustrations consolidating the analytic understanding achieved.Comment: 21 pages, 8 figure
Nearly non-scattering electromagnetic wave set and its application

For any inhomogeneous compactly supported electromagnetic (EM) medium, it is
shown that there exists an infinite set of linearly independent electromagnetic
waves which generate nearly vanishing scattered wave fields. If the
inhomogeneous medium is coated with a layer of properly chosen conducting
medium, then the wave set is generated from the Maxwell-Herglotz approximation
to the interior PEC or PMC eigenfunctions and depends only on the shape of the
inhomogeneous medium. If no such a conducting coating is used, then the wave
set is generated from the Maxwell-Herglotz approximation to the generalised
interior transmission eigenfunctions and depends on both the content and the
shape of the inhomogeneous medium. We characterise the nearly non-scattering
wave sets in both cases with sharp estimates. The results can be used to give a
conceptual design of a novel shadowless lamp. The crucial ingredient is to
properly choose the source of the lamp so that nearly no shadow will be
produced by the surgeons operating under the lamp.Comment: 18 pages, 1 figur
Optimal decay rate for the wave equation on a square with constant
  damping on a strip

We consider the damped wave equation with Dirichlet boundary conditions on
the unit square. We assume the damping to be a characteristic function of a
strip. We prove the exact $t^{-4/3}$-decay rate for the energy of classical
solutions. This answers a question of Anantharaman and L\'eautaud (2014).Comment: 10 pages. In version 2 we corrected a minor mistake in the
  formulation of Theorem 1. A slightly extended and revised version of the
  paper will be published in the "Zeitschrift f\"ur angewandte Mathematik und
  Physik" in April 2017 (online 15th of February
Multiplicity and concentration behavior of solutions to the critical
  Kirchhoff type problem

In this paper, we study the multiplicity and concentration of the positive
solutions to the following critical Kirchhoff type problem: \begin{equation*}
  -\left(\varepsilon^2 a+\varepsilon b\int_{\R^3}|\nabla u|^2\mathrm{d}
x\right)\Delta u + V(x) u = f(u)+u^5\ \ {\rm in } \ \ \R^3, \end{equation*}
where $\varepsilon$ is a small positive parameter, $a$, $b$ are positive
constants, $V \in C(\mathbb{R}^3)$ is a positive potential, $f \in C^1(\R^+,
\R)$ is a subcritical nonlinear term, $u^5$ is a pure critical nonlinearity.
When $\varepsilon>0$ small, we establish the relationship between the number of
positive solutions and the profile of the potential $V$. The exponential decay
at infinity of the solution is also obtained. In particular, we show that each
solution concentrates around a local strict minima of $V$ as $\varepsilon
\rightarrow 0$
On the propagation of transient waves in a viscoelastic Bessel medium

In this paper we discuss the uniaxial propagation of transient waves within a
semi-infinite viscoelastic Bessel medium. First, we provide the analytic
expression for the response function of the material as we approach the
wave-front. To do so, we take profit of a revisited version of the so called
Buchen-Mainardi algorithm. Secondly, we provide an analytic expression for the
long time behavior of the response function of the material. This result is
obtained by means of the Tauberian theorems for the Laplace transform. Finally,
we relate the obtained results to a peculiar model for fluid-filled elastic
tubes.Comment: 14 pages, 4 figure
On D.Y. Gao and X. Lu paper "On the extrema of a nonconvex functional
  with double-well potential in 1D"

The aim of this paper is to discuss the main result in the paper by D.Y. Gao
and X. Lu [On the extrema of a nonconvex functional with double-well potential
in 1D, Z. Angew. Math. Phys. (2016) 67:62]. More precisely we provide a
detailed study of the problem considered in that paper, pointing out the
importance of the norm on the space $C^{1}[a,b]$; because no norm (topology) is
mentioned on $C^{1}[a,b]$ we look at it as being a subspace of $W^{1,p}(a,b)$
for $p\in [1,\infty]$ endowed with its usual norm. We show that the objective
function has not local extrema with the mentioned constraints for $p\in [1,4)$,
and has (up to an additive constant) only a local maximizer for $p=\infty$,
unlike the conclusion of the main result of the discussed paper where it is
mentioned that there are (up to additive constants) two local minimizers and a
local maximizer. We also show that the same conclusions are valid for the
similar problem treated in the preprint by X. Lu and D.Y. Gao [On the extrema
of a nonconvex functional with double-well potential in higher dimensions,
arXiv:1607.03995].Comment: 12 pages; in this version we added the forgotten condition $F(x) \ne
  0$ for $x\in (a,b)$ on page
Global well-posedness of three-dimensional Navier-Stokes equations with
  partial viscosity under helical symmetry

In this paper, we investigate the global well-posedness of three-dimensional
Navier-Stokes equations with horizontal viscosity under a special symmetric
structure: helical symmetry. More precisely, by a revised Ladyzhenskaya-type
inequality and utilizing the behavior of helical flow, we prove the global
existence and uniqueness of weak and strong solution to the three-dimensional
helical flows. Our result reveals that for the issue of global well-posedness
of the viscous helical fluids, the horizontal viscosity plays the important
role. To some extent, our work can be seen as a generalization of the result by
Mahalov-Titi-Leibovich [Arch. Ration. Mech. Anal. 112 (1990), no. 3, 193-222].Comment: 16 page
Global solutions to the electrodynamic two-body problem on a straight
  line

The classical electrodynamic two-body problem has been a long standing open
problem in mathematics. For motion constrained to the straight line, the
interaction is similar to that of the two-body problem of classical
gravitation. The additional complication is the presence of unbounded
state-dependent delays in the Coulomb forces due to the finiteness of the speed
of light. This circumstance renders the notion of local solutions meaningless,
and therefore, straight-forward ODE techniques can not be applied. Here, we
study the time-symmetric case, i.e., the Fokker-Schwarzschild-Tetrode (FST)
equations, comprising both advanced and retarded delays. We extend the
technique developed in \cite{DirkGuenter}, where existence of FST solutions was
proven on the half-line, to ensure global existence -- a result that had been
obtained by Bauer \cite{Bauer} in 1997. Due to the novel technique, the
presented proof is shorter and more transparent but also relies on the idea to
employ asymptotic data to characterize solutions
Trapped modes in zigzag graphene nanoribbons

We study a scattering on an ultra-low potential in zigzag graphene
nanoribbon. Using mathematical framework based on the continuous Dirac model
and augumented scattering matrix, we derive a condition for the existence of a
trapped mode. We consider the threshold energies where the continuous spectrum
changes its multiplicity and show that the trapped modes may appear for
energies slightly less than a threshold and its multiplicity does not exceeds
one. We prove that trapped modes do not appear outside the threshold, provided
the potential is sufficiently small
The relaxed-polar mechanism of locally optimal Cosserat rotations for an
  idealized nanoindentation and comparison with 3D-EBSD experiments

The rotation ${\rm polar}(F) \in {\rm SO}(3)$ arises as the unique orthogonal
factor of the right polar decomposition $F = {\rm polar}(F) \cdot U$ of a given
invertible matrix $F \in {\rm GL}^+(3)$. In the context of nonlinear elasticity
Grioli (1940) discovered a geometric variational characterization of ${\rm
polar}(F)$ as a unique energy-minimizing rotation. In preceding works, we have
analyzed a generalization of Grioli's variational approach with weights
(material parameters) $\mu > 0$ and $\mu_c \geq 0$ (Grioli: $\mu = \mu_c$). The
energy subject to minimization coincides with the Cosserat shear-stretch
contribution arising in any geometrically nonlinear, isotropic and quadratic
Cosserat continuum model formulated in the deformation gradient field $F :=
\nabla\varphi: \Omega \to {\rm GL}^+(3)$ and the microrotation field $R: \Omega
\to {\rm SO}(3)$. The corresponding set of non-classical energy-minimizing
rotations $$
  {\rm rpolar}^\pm_{\mu,\mu_c}(F) := \substack{{\rm argmin}\\ R\,\in\,{\rm
SO(3)}}
  \Big\{ W_{\mu, \mu_c}(R\,;F) := \mu\, || {\rm sym}(R^TF - 1)||^2 + \mu_c\,
||{\rm skew}(R^TF - 1)||^2 \Big\} $$ represents a new relaxed-polar mechanism.
Our goal is to motivate this mechanism by presenting it in a relevant setting.
To this end, we explicitly construct a deformation mapping $\varphi_{\rm nano}$
which models an idealized nanoindentation and compare the corresponding optimal
rotation patterns ${\rm rpolar}^\pm_{1,0}(F_{\rm nano})$ with experimentally
obtained 3D-EBSD measurements of the disorientation angle of lattice rotations
due to a nanoindentation in solid copper. We observe that the non-classical
relaxed-polar mechanism can produce interesting counter-rotations. A possible
link between Cosserat theory and finite multiplicative plasticity theory on
small scales is also explored.Comment: 28 pages, 11 figure
On one-dimensional compressible Navier-Stokes equations for a reacting
  mixture in unbounded domains

In this paper we consider the one-dimensional Navier-Stokes system for a
heat-conducting, compressible reacting mixture which describes the dynamic
combustion of fluids of mixed kinds on unbounded domains. This model has been
discussed on bounded domains by Chen (SIAM Jour. Math. Anal., 23 (1992),
609--634) and Chen-Hoff-Trivisa (Arch. Rat. Mech. Anal. 166 (2003), 321--358)
among others, in which the reaction rate function is a discontinuous function
obeying the Arrhenius Law. We prove the global existence of weak solutions to
this model on one-dimensional unbounded domains with large initial data in
$H^1$. Moreover, the large-time behaviour of the weak solution is identified
and proved. In particular, the uniform-in-time bounds for the temperature and
specific volume have been established via energy estimates. For this purpose we
utilise techniques developed by Kazhikhov and coauthors ({\it cf.} Siber. Math.
Jour. 23 (1982), 44--49; Jour. Appl. Math. Mech., 41 (1977), 273--282), as well
as a crucial estimate in the recent work by Li-Liang (Arch. Rat. Mech. Anal.
220 (2016), 1195--1208). Several new estimates are also established, in order
to treat the unbounded domain and the reacting terms.Comment: 22 page