pinecone/core-2020-05-10-deduplication huggingface dataset:
Chemotaxis: from kinetic equations to aggregate dynamics

26 pagesInternational audienceThe hydrodynamic limit for a kinetic model of chemotaxis is investigated. The limit equation is a non local conservation law, for which finite time blow-up occurs, giving rise to measure-valued solutions and discontinuous velocities. An adaptation of the notion of duality solutions, introduced for linear equations with discontinuous coefficients, leads to an existence result. Uniqueness is obtained through a precise definition of the nonlinear flux as well as the complete dynamics of aggregates, i.e. combinations of Dirac masses. Finally a particle method is used to build an adapted numerical scheme
A homogenization approach for the motion of motor proteins

International audienceWe consider the asymptotic behavior of an evolving weakly coupled Fokker-Planck system of two equations set in a periodic environment. The magnitudes of the diffusion and the coupling are respectively proportional and inversely proportional to the size of the period. We prove that, as the period tends to zero, the solutions of the system either propagate (concentrate) with a fixed constant velocity (determined by the data) or do not move at all. The system arises in the modeling of motor proteins which can take two different states. Our result implies that, in the limit, the molecules either move along a filament with a fixed direction and constant speed or remain immobile
On the supercritical KDV equation with time-oscillating nonlinearity

For the initial value problem (IVP) associated to the generalized
Korteweg-de Vries (gKdV) equation with supercritical nonlinearity,
\begin{equation*}
u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5,
\end{equation*}
numerical evidence [Bona J.L., Dougalis V.A., Karakashian O.A., McKinney W.R.: Conservative, high-order numerical schemes for the generalized Korteweg–de Vries equation. Philos. Trans. Roy. Soc. London Ser. A 351, 107–164 (1995) ] shows that, there are initial data $\phi\in
H^1(\mathbb{R})$ such that the corresponding solution may blow-up in finite time.
Also, with the evidence from numerical simulation [Abdullaev F.K., Caputo J.G., Kraenkel R.A., Malomed B.A.: Controlling collapse in Bose–Einstein condensates by temporal modulation of the scattering length. Phys. Rev. A 67, 012605 (2003)  and Konotop V.V., Pacciani P.: Collapse of solutions of the nonlinear Schrödinger equation with a time dependent nonlinearity: application to the Bose–Einstein condensates. Phys. Rev. Lett. 94, 240405 (2005) ], it has been
claimed that a periodic time dependent coefficient in the nonlinearity would disturb the
blow-up solution, either accelerating or delaying it.
In this work, we investigate the IVP associated to the gKdV equation
\begin{equation*}
u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^{k+1}) =0,
\end{equation*}
where $g$ is a periodic function and $k\geq 5$ is an integer. We prove that, for given
initial data $\phi \in H^1(\mathbb{R})$, as $|\omega|\to \infty$, the solution $u_{\omega}$
converges to the solution $U$ of the initial value problem associated to
\begin{equation*}
U_{t}+\partial_x^3U+m(g)\partial_x(U^{k+1}) =0,
\end{equation*}
with the same initial data, where $m(g)$ is the average of the periodic function $g$.
Moreover, if the solution $U$ is global and satisfies $\|U\|_{L_x^5L_t^{10}}<\infty$, then
we prove that the solution $u_{\omega}$ is also global provided $|\omega|$ is
sufficiently large.M. P. was partially supported by the Research Center of Mathematics of the University of Minho, Portugal through the FCT Pluriannual Funding Program, and through the project PTDC/MAT/109844/2009, and M. S. was partially supported by FAPESP Brazil
Threshold phenomena for symmetric decreasing solutions of
  reaction-diffusion equations

We study the long time behavior of solutions of the Cauchy problem for
nonlinear reaction-diffusion equations in one space dimension with the
nonlinearity of bistable, ignition or monostable type. We prove a one-to-one
relation between the long time behavior of the solution and the limit value of
its energy for symmetric decreasing initial data in $L^2$ under minimal
assumptions on the nonlinearities. The obtained relation allows to establish
sharp threshold results between propagation and extinction for monotone
families of initial data in the considered general setting
On the stability of travelling waves with vorticity obtained by minimization

We modify the approach of Burton and Toland Comm. Pure Appl. Math. (2011) to show the existence of periodic surface water waves with vorticity in order that it becomes suited to a stability analysis. This is achieved by enlarging the function space to a class of stream functions that do not correspond necessarily to travelling profiles. In particular, for smooth profiles and smooth stream functions, the normal component of the velocity field at the free boundary is not required a priori to vanish in some Galilean coordinate system. Travelling periodic waves are obtained by a direct minimisation of a functional that corresponds to the total energy and that is therefore preserved by the time-dependent evolutionary problem (this minimisation appears in Burton and Toland after a first maximisation). In addition, we not only use the circulation along the upper boundary as a constraint, but also the total horizontal impulse (the velocity becoming a Lagrange multiplier). This allows us to preclude parallel flows by choosing appropriately the values of these two constraints and the sign of the vorticity. By stability, we mean conditional energetic stability of the set of minimizers as a whole, the perturbations being spatially periodic of given period
On The Fu\v{c}ik Spectrum Of Non-Local Elliptic Operators

In this article, we study the Fu\v{c}ik spectrum of fractional Laplace
operator which is defined as the set of all $(\al,\ba)\in \mb
  R^2$ such that
  \begin{equation*}
  \quad \left. \begin{array}{lr}
  \quad (-\De)^s u = \al u^{+} - \ba u^{-} \; \text{in}\; \Om
  \quad \quad \quad \quad u = 0 \; \mbox{in}\; \mb R^n \setminus\Om.\\
\end{array} \quad \right\} \end{equation*} has a non-trivial solution $u$,
where $\Om$ is a bounded domain in $\mb R^n$ with Lipschitz boundary, $n>2s$,
$s\in(0,1)$. The existence of a first nontrivial curve $\mc C$ of this
spectrum, some properties of this curve $\mc C$, e.g. Lipschitz continuous,
strictly decreasing and asymptotic behavior are studied in this article. A
variational characterization of second eigenvalue of the fractional eigenvalue
problem is also obtained. At the end, we study a nonresonance problem with
respect to Fu\v{c}ik spectrum.Comment: 22 pages in NoDEA: Nonlinear differential equations and applications,
  201
Local Strong Solutions for the Non-Linear Thermoelastic Plate Equation
  on Rectangular Domains in $L^p$-Spaces

We consider the non-linear thermoelastic plate equation in rectangular
domains $\Omega$. More precisely, $\Omega$ is considered to be given as the
Cartesian product of whole or half spaces and a cube. First the linearized
equation is treated as an abstract Cauchy problem in $L^p$-spaces. We take
advantage of the structure of $\Omega$ and apply operator-valued Fourier
multiplier results to infer an $\mathcal R$-bounded $\mathcal
H^\infty$-calculus. With the help of maximal $L^p$-regularity existence and
uniqueness of local real-analytic strong solutions together with analytic
dependency on the data is shown.Comment: 16 pages; final versio
Controllability for systems governed by semilinear evolution equations without compactness

We study the controllability for a class of semilinear differential inclusions in Banach spaces. Since we assume the regularity of the nonlinear part with respect to the weak topology, we do not require the compactness of the evolution operator generated by the linear part. As well we are not posing any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. We are considering the usual assumption on the controllability of the associated linear problem. Notice that, in infinite dimensional spaces, the above mentioned compactness of\ud
the evolution operator and linear controllability condition are in contradiction with each other. We suppose that the nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. This regularity setting allows us to solve controllability problem under various growth conditions. As application, a controllability result for hyperbolic integro-differential equations and inclusions is obtained. In particular, we consider controllability of a system arising in a model of nonlocal spatial population dispersal and a system governed by the second order one-dimensional telegraph equation
Nonzero radial solutions for a class of elliptic systems with nonlocal
  BCs on annular domains

We provide new results on the existence, non-existence, localization and
multiplicity of nontrivial solutions for systems of Hammerstein integral
equations. Some of the criteria involve a comparison with the spectral radii of
some associated linear operators. We apply our results to prove the existence
of multiple nonzero radial solutions for some systems of elliptic boundary
value problems subject to nonlocal boundary conditions. Our approach is
topological and relies on the classical fixed point index. We present an
example to illustrate our theory.Comment: 25 pages. arXiv admin note: text overlap with arXiv:1404.139
On the viscosity solutions to Trudinger's equation

We study the existence of positive viscosity solutions to Trudinger's
equation for cylindrical domains $\Omega\times[0, T)$, where $\Omega\subset
\mathbb{R}^n,\;n\ge 2,$ is a bounded domain, $T>0$ and $2\leq p<\infty$. We
show existence for general domains $\Omega,$ when $n<p<\infty$. For $2\leq
p\leq n$, we prove existence for domains $\Omega$ that satisfy a uniform outer
ball condition. We achieve this by constructing suitable sub-solutions and
super-solutions and applying Perron's method.Comment: 25 pages, 2 figure
Propagation of singularities for semilinear Schr\"odinger equations

We study the propagation of singularities for semilinear Schrodinger
equations with quadratic Hamiltonians, in particular for the semilinear
harmonic oscillator. We show that the propagation still occurs along the flow
the Hamiltonian flow, but for Sobolev regularities in a certain range and
provided the notion of Sobolev-wave front set is conveniently modified. The
proof makes use of a weighted version of the paradifferential calculus, adapted
to our situation. The results can be regarded as the Schrodinger counterpart of
those known for semilinear hyperbolic equations, which hold with the usual wave
front set.Comment: 16 page
Renormalized solutions of semilinear equations involving measure data
  and operator corresponding to Dirichlet form

We generalize the notion of renormalized solution to semilinear elliptic and
parabolic equations involving operator associated with general (possibly
nonlocal) regular Dirichlet form and smooth measure on the right-hand side. We
show that under mild integrability assumption on the data a quasi-continuous
function $u$ is a renormalized solution to an elliptic (or parabolic) equation
in the sense of our definition iff $u$ is its probabilistic solution, i.e. $u$
can be represented by a suitable nonlinear Feynman-Kac formula. This implies in
particular that for a broad class of local and nonlocal semilinear equations
there exists a unique renormalized solution
The ergodic problem for some subelliptic operators with unbounded coefficients

article 47International audienceWe study existence and uniqueness of the invariant measure for a stochastic process with degenerate diffusion, whose infinitesimal generator is a linear subelliptic operator in the whole space R N with coefficients that may be unbounded. Such a measure together with a Liouville-type theorem will play a crucial role in two applications: the ergodic problem studied through stationary problems with vanishing discount and the long time behavior of the solution to a parabolic Cauchy problem. In both cases, the constants will be characterized in terms of the invariant measure
Higher order asymptotic expansions to the solutions for a nonlinear
  damped wave equation

We study the Cauchy problem for a nonlinear damped wave equation. Under
suitable assumptions for the nonlinearity and the initial data, we obtain the
global solution which satisfies weighted $L^1$ and $L^\infty$ estimates.
Furthermore, we establish the higher order asymptotic expansion of the
solution. This means that we construct the nonlinear approximation of the
global solution with respect to the weight of the data. Our proof is based on
the approximation formula of the linear solution, which is given in [36], and
the nonlinear approximation theory for a nonlinear parabolic equation developed
by [18]
Compactness and existence results in weighted Sobolev spaces of radial
  functions. Part II: Existence

We prove existence and multiplicity results for finite energy solutions to
the nonlinear elliptic equation \[ -\triangle u+V\left( \left| x\right| \right)
u=g\left( \left| x\right| ,u\right) \quad \textrm{in }\Omega \subseteq
\mathbb{R}^{N},\ N\geq 3, \] where $\Omega $ is a radial domain (bounded or
unbounded) and $u$ satisfies $u=0$ on $\partial \Omega $ if $\Omega \neq
\mathbb{R}^{N}$ and $u\rightarrow 0$ as $\left| x\right| \rightarrow \infty $
if $\Omega $ is unbounded. The potential $V$ may be vanishing or unbounded at
zero or at infinity and the nonlinearity $g$ may be superlinear or sublinear.
If $g$ is sublinear, the case with $g\left( \left| \cdot \right| ,0\right) \neq
0$ is also considered.Comment: 29 pages, 8 figure
Choquard equations under confining external potentials

We consider the nonlinear Choquard equation $$ -\Delta u+V u=(I_\alpha \ast
\vert u\vert ^p)\vert u\vert ^{p-2}u \qquad \text{ in } \mathbb{R}^N $$ where
$N\geq 1$, $I_\alpha$ is the Riesz potential integral operator of order $\alpha
\in (0, N)$ and $p > 1$. If the potential $ V \in C (\mathbb{R}^N; [0,+\infty))
$ satisfies the confining condition $$ \liminf\limits_{\vert x\vert \to
+\infty}\frac{V(x)}{1+\vert x\vert ^{\frac{N+\alpha}{p}-N}}=+\infty, $$ and
$\frac{1}{p} > \frac{N - 2}{N + \alpha}$, we show the existence of a
groundstate, of an infinite sequence of solutions of unbounded energy and, when
$p \ge 2$ the existence of least energy nodal solution. The constructions are
based on suitable weighted compact embedding theorems. The growth assumption is
sharp in view of a Poho\v{z}aev identity that we establish.Comment: 21 page
The Cauchy problem for the nonlinear damped wave equation with slowly
  decaying data

We study the Cauchy problem for the nonlinear damped wave equation and
establish the large data local well-posedness and small data global
well-posedness with slowly decaying initial data. We also prove that the
asymptotic profile of the global solution is given by a solution of the
corresponding parabolic problem, which shows that the solution of the damped
wave equation has the diffusion phenomena. Moreover, we show blow-up of
solution and give the estimate of the lifespan for a subcritical nonlinearity.
In particular, we determine the critical exponent for any space dimension.Comment: 43 pages. Theorem 1.3 is improved, some errors are corrected and
  references are update
Decay rates for the quadratic and super-quadratic tilt-excess of
  integral varifolds

This paper concerns integral varifolds of arbitrary dimension in an open
subset of Euclidean space satisfying integrability conditions on their first
variation. Firstly, the study of pointwise power decay rates almost everywhere
of the quadratic tilt-excess is completed by establishing the precise decay
rate for two-dimensional integral varifolds of locally bounded first variation.
In order to obtain the exact decay rate, a coercive estimate involving a
height-excess quantity measured in Orlicz spaces is established. Moreover,
counter-examples to pointwise power decay rates almost everywhere of the
super-quadratic tilt-excess are obtained. These examples are optimal in terms
of the dimension of the varifold and the exponent of the integrability
condition in most cases, for example if the varifold is not two-dimensional.
These examples also demonstrate that within the scale of Lebesgue spaces no
local higher integrability of the second fundamental form, of an at least
two-dimensional curvature varifold, may be deduced from boundedness of its
generalised mean curvature vector.
  Amongst the tools are Cartesian products of curvature varifolds.Comment: mainly extended the overview section; updated reference
Gradient estimates for mean curvature flow with Neumann boundary
  conditions

We study the mean curvature flow of graphs both with Neumann boundary
conditions and transport terms. We derive boundary gradient estimates for the
mean curvature flow. As an application, the existence of the mean curvature
flow of graphs is presented. A key argument is a boundary monotonicity formula
of a Huisken type derived using reflected backward heat kernels. Furthermore,
we provide regularity conditions for the transport terms.Comment: 21 pages, 3 figures, Accepted for publication in "NoDEA. Nonlinear
  Differential Equations and Applications
Asymptotic Results for Solutions of a weighted p-Laplacian evolution
  Equation with Neumann Boundary Conditions

The purpose of this paper is to investigate the time behavior of the solution
of a weighted $p$-Laplacian evolution equation, given by \begin{align}
\label{eveq} \begin{cases} u_{t} = \text{div} \left(\gamma |\nabla
u|^{p-2}\nabla u \right) & \text{on} (0,\infty)\times S, \\ \gamma|\nabla
u|^{p-2}\nabla u\cdot\eta=0 & \text{on} (0,\infty)\times \partial S, \\
u(0,\cdot)=u_{0} & \text{on} S,\end{cases} \end{align} where $n \in
\mathbb{N}\setminus \{1\}$, $p \in (1,\infty)\setminus \{2\}$, $S\subseteq
\mathbb{R}^{n}$ is an open, bounded and connected set of class $C^{1}$, $\eta$
is the unit outer normal on $\partial S $, and $\gamma: S \rightarrow
(0,\infty)$ is a bounded function which can be extended to an
$A_{p}$-Muckenhoupt weight on $\mathbb{R}^{n}$. It will be proven that the
solution converges in $L^{1}(S)$ to the average of the initial value $u_{0} \in
L^{1}(S)$. Moreover, a conservation of mass principle, an extinction principle
and a decay rate for the solution will be derived.Comment: The final paper is published in NoDEA (August 2017 - Volume 24 Issue
  4) at http://link.springer.co
Stochastic Allen-Cahn equation with mobility

We introduce a class of stochastic Allen-Cahn equations with a mobility
coefficient and colored noise. For initial data with finite free energy, we
analyze the corresponding Cauchy problem on the $d$-dimensional torus in the
time interval $[0,T]$. Assuming that $d\le 3$ and that the potential has
quartic growth, we prove existence and uniqueness of the solution as a process
$u$ in $L^2$ with continuous paths, satisfying almost surely the regularity
properties $u\in C([0,T]; H^1)$ and $u\in L^2([0,T];H^2)$.Comment: 34 page
Fractional Choquard Equation with Critical Nonlinearities

In this article, we study the Brezis-Nirenberg type problem of nonlinear
Choquard equation involving a fractional Laplacian \[ (-\De)^s u = \left(
\int_{\Om}\frac{|u|^{2^*_{\mu,s}}}{|x-y|^{\mu}}\mathrm{d}y
\right)|u|^{2^*_{\mu,s}-2}u +\la u \; \text{in } \Om,\] where $\Om $ is a
bounded domain in $\mathbb R^n$ with Lipschitz boundary, $\la $ is a real
parameter, $s \in (0,1)$, $n >2s$ and $2^*_{\mu,s}= (2n-\mu)/(n-2s)$ is the
critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. We
obtain some existence, multiplicity, regularity and nonexistence results for
solution of the above equation using variational methods.Comment: 32 pages. arXiv admin note: text overlap with arXiv:1604.00826 by
  other author
The prescribed mean curvature equation in weakly regular domains

We show that the characterization of existence and uniqueness up to vertical
translations of solutions to the prescribed mean curvature equation, originally
proved by Giusti in the smooth case, holds true for domains satisfying very
mild regularity assumptions. Our results apply in particular to the
non-parametric solutions of the capillary problem for perfectly wetting fluids
in zero gravity. Among the essential tools used in the proofs, we mention a
\textit{generalized Gauss-Green theorem} based on the construction of the weak
normal trace of a vector field with bounded divergence, in the spirit of
classical results due to Anzellotti, and a \textit{weak Young's law} for
$(\Lambda,r_{0})$-minimizers of the perimeter.Comment: 23 pages, 1 figure --- The results on the weak normal trace of vector
  fields have been now extended and moved in a self-contained paper available
  at: arXiv:1708.0139
Dual Variational Methods for a nonlinear Helmholtz system

This paper considers a pair of coupled nonlinear Helmholtz equations
\begin{align*}
  -\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2}
\right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*}
  -\Delta v - \nu v = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2}
\right)|v|^{\frac{p}{2} - 2}v \end{align*} on $\mathbb{R}^N$ where
$\frac{2(N+1)}{N-1} < p < 2^\ast$. The existence of nontrivial strong solutions
in $W^{2, p}(\mathbb{R}^N)$ is established using dual variational methods.
  The focus lies on necessary and sufficient conditions on the parameters
deciding whether or not both components of such solutions are nontrivial.Comment: Published version. Contains minor revisions: Quote added,
  explanations on p.12 concerning F_{\mu\nu} = \infty, correction of exponent
  on p.1
A universal construction for moduli spaces of decorated vector bundles
  over curves

Let $X$ be a smooth projective curve over the complex numbers. To every
representation $\rho\colon \GL(r)\lra \GL(V)$ of the complex general linear
group on the finite dimensional complex vector space $V$ which satisfies the
assumption that there be an integer $\alpha$ with $\rho(z \id_{\C^r})=z^\alpha
\id_V$ for all $z\in\C^*$ we associate the problem of classifying triples
$(E,L,\phi)$ where $E$ is a vector bundle of rank $r$ on $X$, $L$ is a line
bundle on $X$, and $\phi\colon E_\rho\lra L$ is a non trivial homomorphism.
Here, $E_\rho$ is the vector bundle of rank $\dim V$ associated to $E$ via
$\rho$. If we take, for example, the standard representation of $\GL(r)$ on
$\C^r$ we have to classify triples $(E,L,\phi)$ consisting of $E$ as before and
a non-zero homomorphism $\phi\colon E\lra L$ which includes the so-called
Bradlow pairs. For the representation of $\GL(r)$ on $S^2\C^3$ we find the
conic bundles of Gomez and Sols. In the present paper, we will formulate a
general semistability concept for the above triples which depends on a rational
parameter $\delta$ and establish the existence of moduli spaces of
$\delta$-(semi)stable triples of fixed topological type. The notion of
semistability mimics the Hilbert-Mumford criterion for $SL(r)$ which is the
main reason that such a general approach becomes feasible. In the known
examples (the above, Higgs bundles, extension pairs, oriented framed bundles)
we show how to recover the "usual" semistability concept. This process of
simplification can also be formalized. Altogether, our results provide a
unifying construction for the moduli spaces of most decorated vector bundle
problems together with an automatism for finding the right notion of
semistability and should therefore be of some interest.Comment: Final Version (To appear in Transformation Groups); V2: Example 3.7
  corrected, other minor modifications; V3: Notion of polystability corrected,
  other minor modification