sharp estimates for solutions of multi bubbles in compact riemann surfaces in this paper we consider a sequence of multibubble solutions u k of the equation 0 1 0 u k k he uk f m he u k d 1 0 in m where h is a c 2 positive function in a compact riemann surface m and k is a constant satisfying lim k k 8m for some positive integer m 1 we prove among other things that k 8m 2 m m j 1h 1 p k j 0 log h p k j 8m 2k p k j k j e k j o e k j where p k j are centers of the bubbles of u k and k j are the local maxima of u k after adding a constant this yields a uniform bound of solutions as k converges to 8m from below provided that 0 log h p k j 8m 2k p k j 0 it generalizes a previous result due to ding jost li and wang 18 and nolasco and tarantello 31 which says that any sequence of minimizers u k is uniformly bounded if k 0 for any maximum point p of the sum of 2 log h and the regular part of the green function where k is the gaussian curvature of m the analytic work of this paper is the first step toward computing the topological degree of 0 1 which was initiated by li 24

on the global well posedness theory for a class of pde models for criminal activity abstract we study a class of reaction advection diffusion system of partial differential equations which can be taken as basic models for criminal activity this class of models are based on routine activity theory and other theories such as the repeat and near repeat victimization effect and were first introduced in short et al 2008 11 in these models the criminal density is advected by a velocity field that depends on a scalar field which measures the appeal to commit a crime we refer to this scalar field as the attractiveness field we prove local well posedness of solutions for the general class of models furthermore we prove global well posedness of solutions to a fully parabolic system with a velocity field that depends logarithmically on the attractiveness field our final result is the global well posedness of solutions the fully parabolic system with velocity field that depends linearly on the attractiveness field for small initial mass

variational and projection methods for the volume constraint in finite deformation elasto plasticity this paper focuses on the treatment of volume constraints which in the context of elasto plasticity typically arise as a result of assuming volume preserving plastic flow projection methods based on the modification of the discrete gradient operator b often proposed on an ad hoc basis are systematically obtained in the variational context furnished by a three field hu washizu principle the fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume preserving and dilatational parts without relying on rate forms of the weak form of momentum balance this approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient and enables one to exactly enforce the condition of volume preserving plastic flow within the framework proposed in this paper rate forms and incrementally objective algorithms are entirely bypassed

energy pairs in the micropolar continuum in this paper the concept of energy pairs in the micropolar continuum is introduced a brief review of the micropolar continuum theory is presented for using in the subsequent derivations a mathematical lagrangian strain and a wryness tensor for the micropolar continuum are introduced using the first law of thermodynamics and for isothermal processes the power of deformation is obtained and the energy pairs in the eulerian and lagrangian descriptions are defined also the micropolar stress and couple stress tensors which are energy pairs to the micropolar lagrangian strain and wryness measures are determined

maximal solution of the liouville equation in doubly connected domains abstract in this paper we consider the liouville equation u 2 e u 0 with dirichlet boundary conditions in a two dimensional doubly connected domain we show that there exists a simple closed curve such that for a sequence n 0 and a sequence of solutions u n it holds u n log 1 n h where h is a harmonic function in and n 2 log 1 n e u n d x 8 c where c is a constant depending on the conformal class of only

radial solutions for the brezis nirenberg problem involving large nonlinearities abstract let us consider the problem 0 1 u a x u u p in b 1 u 0 in b 1 u 0 on b 1 where b 1 is the unit ball in r n n 3 and a x 0 is a smooth radial function under some suitable assumptions on the regular part of the green function of the operator u n 1 r u a r u we prove the existence of a radial solution to 0 1 for p large enough

aggregation blowup and collapse the abc s of taxis in reinforced random walks in many biological systems movement of an organism occurs in response to a diffusible or otherwise transported signal and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by patlak bull of math biophys 15 1953 pp 311 338 however other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages in these systems one example of which is the myxobacteria the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates davis probab theory related fields 84 1990 pp 203 229 has studied this question for a certain class of random walks and here we extend this analysis to the continuum limit of such walks we first derive several general classes of partial differential equations that depend on how the movement rules are

existence and concentration of positive solutions for a system of coupled saturable schrodinger equations abstract it is considered a saturable system of coupled schrodinger equations with asymptotically linear nonlinearities the bound states solutions with finite energy are proved to exist and have a concentration behavior under hypotheses involving the coupling and saturation parameters the proof is based on variational methods

remarks on rate constitutive equations for finite deformation problems computational implications it is explicitly shown that if the spatial elasticity tensor of an elastic material is taken as isotropic for all possible configurations then its coefficients cannot be constants they must depend nontrivially on the jacobian determinant of the deformation gradient moreover the assumption typically made for computational purposes that its coefficients remain constant for all possible configurations is incompatible with elasticity it is further shown that an assumption widely used in the computational literature in the context of finite deformation plasticity namely relating an objective stress rate to the rate of deformation tensor through a fourth rank constant isotropic tensor is also incompatible with elasticity thus furnishing an example of an hypoelastic material which is not elastic

statistical mechanics of classical particles with logarithmic interactions the inhomogeneous mean field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions the results apply to various physical models of translation invariant plasmas gravitating systems as well as to planar fluid vortex motion for attractive interactions a critical behavior occurs which can be classified as an extreme case of a second order phase transition to include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic geometric mean inequality the method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions in addition it also provides us with a new proof of the trudinger moser inequality known from differential geometry in its sharp form

bubble tower radial solutions in the slightly supercritical brezis nirenberg problem abstract in this paper we consider the brezis nirenberg problem in dimension n 4 in the supercritical case we prove that if the exponent gets close to n 2 n 2 and if simultaneously the bifurcation parameter tends to zero at the appropriate rate then there are radial solutions which behave like a superposition of bubbles namely solutions of the form j 1 k 1 1 m j 4 n 2 y 2 n 2 2 m j 1 o 1 n n 2 n 2 4 where mj and mj o mj 1 for all j these solutions lie close to turning points to the right of the associated bifurcation diagram

spatio temporal oscillations in the keller segel system with logistic growth abstract the keller segel system with the logistic growth term is discussed from the spatio temporal oscillation point of view this system exhibits two different types of spatio temporal oscillations in certain distinct parameter regimes in this paper we study the difference between the two types of spatio temporal oscillations in particular the characteristic properties of the behaviors become clear in a limiting system when a certain parameter value tends to zero moreover we demonstrate that the onset of one of the spatio temporal oscillatory patterns is an infinite dimensional relaxation oscillation that consists of slow and fast dynamics

large mass boundary condensation patterns in the stationary keller segel system abstract we consider the boundary value problem u u e u in u 0 on where is a bounded smooth domain in r 2 0 and is the inner normal derivative at this problem is equivalent to the stationary keller segel system from chemotaxis we establish the existence of a solution u which exhibits a sharp boundary layer along the entire boundary as 0 these solutions have large mass in the sense that e u log

a system of reaction diffusion equations arising in the theory of reinforced random walks we investigate the properties of solutions of a system of chemotaxis equations arising in the theory of reinforced random walks we show that under some circumstances finite time blow up of solutions is possible in other circumstances the solutions will decay to a spatially constant solution collapse we also give some intuitive arguments which demonstrate the possibility of the existence of aggregation piecewise constant solutions

free vibration of size dependent mindlin microplates based on the modified couple stress theory this paper develops a mindlin microplate model based on the modified couple stress theory for the free vibration analysis of microplates this non classical plate model contains an internal material length scale parameter related to the material microstructures and is capable of interpreting the size effect that the classical mindlin plate model is unable to describe the higher order governing equations of motion and boundary conditions are derived using the hamilton principle the p version ritz method is employed to determine the natural frequencies of the microplate with different boundary conditions a detailed parametric study is conducted to study the influences of the length scale parameter side to thickness ratio and aspect ratio on the free vibration characteristics of the microplate it is found that the size effect is significant when the thickness of microplate is close to the material length scale parameter

material symmetry group of the non linear polar elastic continuum we extend the material symmetry group of the non linear polar elastic continuum by taking into account microstructure curvature tensors as well as different transformation properties of polar and axial tensors the group consists of an ordered triple of tensors which makes the strain energy density of polar elastic continuum invariant under change of reference placement an analog of the noll rule is established four simple specific cases of the group with corresponding reduced forms of the strain energy density are discussed definitions of polar elastic fluids solids liquid crystals and subfluids are given in terms of members of the symmetry group within polar elastic solids we discuss in more detail isotropic hemitropic cubic symmetric transversely isotropic and orthotropic materials and give explicitly corresponding reduced representations of the strain energy density for physically linear polar elastic solids when the density becomes a quadratic function of strain measures reduced representations of the density are established for monoclinic orthotropic cubic symmetric hemitropic and isotropic materials in terms of appropriate joint scalar invariants of stretch wryness and undeformed structure curvature tensors

on a time dependent bio reactor model with chemotaxis long time dynamics of solutions to a strongly coupled system of parabolic equations modeling the competition in bio reactors with chemotaxis will be studied if the parameters of the system are periodic sufficient conditions for positive periodic solutions will be derived in terms of parabolic eigenvalue problems

nonexistence results for elliptic problems with supercritical nonlinearity in nontrivial domains let be a bounded domain in rn with n 3 and assume p 2n n 2 we study problems of the type u u p 2u 0 in u 0 on we prove that for suitable p 2n n 2 there exists some bounded domain with nontrivial topology in the sense of bahri and coron such that the above problem has only the solution u 0

formal and analytical integrability of the bianchi ix system in this paper we provide a complete description of the first integrals of the classical bianchi ix system that can be described by a general class of formal power series as a corollary we also obtain a complete description of some of its analytic first integrals in a neighborhood of the origin in particular we prove that the system is not completely integrable by analytic first integrals

asymptotic behavior of blow up solutions to a class of prescribing gauss curvature equations the liouville equation u v x eu 0 has many applications in science and mathematics one of the interesting features of solutions is their blow up behavior a topic for which many results have been obtained in this paper we give a simple approach to obtaining a precise asymptotic expansion of a sequence of blow up solutions to the liouville equation in a bounded domain r2

micropolar hyper elastoplasticity constitutive model consistent linearization and simulation of 3d scale effects summary a computational model for micropolar hyperelastic based finite elastoplasticity that incorporates isotropic hardening is developed the basic concepts of the non linear micropolar kinematic framework are reviewed and a thermodynamically consistent constitutive model that features neo hooke type elasticity and generalized von mises plasticity is described the integration of the constitutive initial value problem is carried out by means of an elastic predictor plastic corrector algorithm which retains plastic incompressibility the solution procedure is developed carefully and described in detail the consistent material tangent is derived the micropolar constitutive model is implemented in an implicit finite element framework the numerical example of a notched cylindrical bar subjected to large axial displacements and large twist angles is presented the results of the finite element simulations demonstrate i that the methodology is capable of capturing the size effect in three dimensional elastoplastic solids in the finite strain regime ii that the formulation possesses a regularizing effect in the presence of strain localization and iii that asymptotically quadratic convergence rates of the newton raphson procedure are achieved throughout this paper effort is made to present the developments as a direct extension of standard finite deformation computational plasticity copyright 2012 john wiley sons ltd

localized failure analysis in elastoplastic cosserat continua this paper summarizes our experience with localized failure analysis in micropolar cosserat continua the main intent is to examine the regularization properties of discontinuous bifurcation problems when higher grade micropolar materials that exhibit an internal length scale are introduced thereby the key feature is the endowment of micropolar continua with rotational drilling degrees of freedom in addition to the traditional description of motion in terms of displacements after a brief review of elastoplastic cosserat continua we examine recent findings of localization analysis of discontinuous bifurcation 1 to this end we generalize the strength concept of the mohr envelope condition to non symmetric stress states to analyze the localization condition of discontinuous failure 2 the mohr representation of stress leads to a geometrical condition which is used to determine critical hardening softening moduli and localization directions that characterize the particular mode of failure this geometric interpretation of the localization condition lends itself to a systematic study of regularizing weak discontinuities within elastoplastic cosserat continua for illustration of these theoretical findings numerical failure studies are carried out on a representative volume element made of j2 type elastoplastic cosserat material the computational failure simulations demonstrate the non local character of cosserat continua that depends on an internal length scale and on the degree of non symmetry in the stress and strain states the computational results also illustrate the conversion of the underlying failure mode from mixed mode to mode i type failure of decohesion and separation

stationary solutions of chemotaxis systems the keller segel model is a system of partial differential equations modelling a mutual attraction of amoebae caused by releasing a chemical substance chemotaxis this paper analyzes the stationary solutions of the system with general nonlinearities via bifurcation techniques and gives a criterion for bifurcation of stable nonhomogeneous aggregation patterns examples are discussed with various kinds of nonlinearities modelling the sensitivity of the chemotaxis response

stability results for a diffusion equation with functional drift approximating a chemotaxis model a hyperbolic parabolic chemotaxis system modelling aggre gation of motile cells by production of a diffusible chemoattractant is ap proximated by a scalar diffusion equation for the cell density where the drift term is an explicit functional of the current density profile we prove the unique existence and using the hopf cole transformation the local stability of an equilibrium i e a steady aggregation state we also discuss the limit ing hyperbolic case of vanishing random motility with the formation of shocks describing cell clumps 1 introduction and development of the model in this note we shall de velop and explore the behavior of a simple two speed one dimensional model which describes the motion of chemosensitive cells that is cells whose motions are influ enced by a chemical gradient dp dx where p p x t denotes the concentration of a diffusible substance produced by each cell a well known example is the mor phogenetic aggregation of slime mold ceils 5 which previously has been modelled by keller and segel 6 using a coupled system of two equations for p and the cell density n u x t the so called chemotaxis system for a recent mathematical analysis see 4 12 in most cases as in slime molds however parameter estimations reveal that the diffusion coefficient of the chemoattractant dp 10 7cm2 sec is about ten times larger than the motility of the cells du 10 8 cm2 sec compare for example 3 if also the production rate of the chemoattractant is relatively large then the quasi stationary gradient of the rapidly diffusing substance approx imately determines the chemotactic drift of the cells this finally leads to a scalar diffusion equation for the cell density u of type

pattern formation i the keller segel model abstract we investigate nonlinear dynamics near an unstable constant equilibrium in the classical keller segel model given any general perturbation of magnitude we prove that its nonlinear evolution is dominated by the corresponding linear dynamics along a fixed finite number of fastest growing modes over a time period of ln 1 our result can be interpreted as a rigorous mathematical characterization for early pattern formation in the keller segel model

constitutive equations for micropolar hyper elastic materials in this paper the concept of hyper elasticity in the micropolar continuum theory is investigated the restrictions on the fourth order elasticity tensors are investigated using the representation theorems a general form of constitutive equations for micropolar hyper elastic isotropic materials is presented as some special cases generalizations of the neo hookean and mooney rivlin type materials to the micropolar continuum theory are presented the generalized constitutive equations reduce to those of the microplar linear elasticity theory when the deformations are infinitesimal also updated lagrangian finite element formulations for the micropolar hyper elastic materials are presented considering two planar examples it is shown that an increase in the micropolar parameter results in the reduction of the deformation of the bodies also it is shown that for a specimen with very small dimensions e g in the micron level the micropolar effects are more sensible furthermore it is shown that the influence of the micropolar parameters is dependent not only on the size of the body but also to its geometry and loading conditions for the problems in which the deformation is very close to a homogeneous state the micropolar effects are negligible

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

a special class of stationary flows for two dimensional euler equations a statistical mechanics description we consider the canonical gibbs measure associated to an vortex system in a bounded domain at inverse temperature widetilde beta and prove that in the limitn widetilde beta n n 1 where 8 here denotes the vorticity intensity of each vortex the one particle distribution function n nx x converges to a superposition of solutions of the following mean field equation left begin array 20 c varrho beta x frac e beta psi mathop smallint limits lambda e beta psi delta psi varrho beta in lambda psi partial lambda 0 end array right

maxwellian decay for well balanced approximations of a super characteristic chemotaxis model we focus on the numerical simulation of a one dimensional model of chemotaxis dynamics proposed by greenberg and alt trans amer math soc 300 1987 pp 235 258 in a bounded domain by means of a previously introduced well balanced wb and asymptotic preserving ap scheme l gosse j math anal appl 2011 we are especially interested in studying the decay onto numerical steady states for two reasons 1 conventional upwind schemes have been shown to stabilize onto spurious non maxwellian states with a very big mass flux see e g f guarguaglini et al discrete contin dyn syst ser b 12 2009 pp 39 76 and 2 the initial data lead to a dynamic which is mostly super characteristic in the sense of s jin and m katsoulakis siam j appl math 61 2000 pp 273 292 thus the stability results of gosse do not apply a reflecting boundary condition which is compatible with the well balanced character is presented a mass preservation property is proved and some results on super characteristic relaxation are recalled numerical experiments with coarse computational grids are presented in detail they illustrate the bifurcation diagrams of guarguaglini et al which relate the total initial mass of cells with the time asymptotic values of the chemoattractant substance on each side of the domain it is shown that the wb scheme stabilizes correctly onto zero mass flow rate hence maxwellian steady states which agree with the aforementioned bifurcation diagrams the evolution in time of residues is commented for every considered test case

the principle of linearized stability for quasilinear parabolic evolution equations on donne des conditions pour la stabilite des points d equilibre des equations paraboliques quasilineaires avec des conditions aux limites non lineaires u a t u u u f t u 7b b t u u 0 u t 0 u 0 t 0 t

formal and analytic first integrals of the einstein yang mills equations in this paper we provide a complete description of the first integrals of the classical einstein yang mills equations that can be described by formal series as a corollary we also obtain a complete description of the analytic first integrals in a neighbourhood of the origin

nonnegativity of exact and numerical solutions of some chemotactic models we investigate nonnegativity of exact and numerical solutions to a generalized keller segel model this model includes the so called minimal keller segel model but can cover more general chemistry we use maximum principles and invariant sets to prove that all components of the solution of the generalized model are nonnegative we then derive numerical methods using finite element techniques for the generalized keller segel model adapting the ideas in our proof of nonnegativity of exact solutions to the discrete setting we are able to show nonnegativity of discrete solutions from the numerical methods under certain standard assumptions one of the numerical methods is then applied to the minimal keller segel model recalling known results on the qualitative behavior of this model we are able to choose parameters that yield convergence to a nonhomogeneous stationary solution while proceeding to exhibit these stationary patterns we also demonstrate how naive choices of numerical methods can give physically unrealistic solutions thereby justifying the need to study positivity preserving methods

geometry of spaces with the jacobi metric the generalized maupertuis principle is formulated for systems with the natural lagrangian and an indefinite form of the kinetic energy the generalization is applied to the theory of gravity and cosmology for such systems the metric determined by the kinetic energy form has a lorentz signature the theorem is proved concerning the behavior of trajectories in a neighborhood of the boundary of the region admissible for motion this region is not a smooth manifold but turns out to be a differential space of constant differential dimension this fact allows us to use geometric methods analogous to those elaborated for smooth manifolds it is shown that singularities of the jacobi metric are not dangerous for the motion its trajectories are smooth in the sense of the theory of differential spaces

asymptotic behavior of solutions to a quasilinear nonuniform parabolic system modelling chemotaxis abstract in this paper we study a quasilinear nonuniform parabolic system modelling chemotaxis and taking the volume filling effect into account the results on the existence of a unique global classical solution was obtained in cie lak 2007 4 however the convergence to equilibrium was not considered in that paper in this paper we first obtain the crucial uniform boundedness of the solution then with the help of a suitable non smooth simon ojasiewicz approach we obtain the results on convergence to equilibrium and the decay rate

the last remake of the mixmaster universe model we review the existing evidence on the non integrability of the mixmaster universe model we show how a local painleve analysis can be used to study the possible existence of essential singularities in agreement with recent studies we find that the mixmaster model possesses critical essential singularities the multivalued character of which is incompatible with integrability our analysis is complemented by a numerical study of the spectrum of the stretching numbers of the system we show that the zero energy case is characterized by a vanishing maximal lyapunov characteristic number

asymptotic behaviour of the kazdan warner solution in the annulus abstract in this paper we study the asymptotic behaviour as p of the radial solution of the problem u u p in u 0 in u 0 on where is an annulus of r n n 2

global bifurcation of steady state solutions in this paper we study the bifurcation of steady state solutions of a reaction diffusion equation in one space variable the steady state solutions satisfy the equation a f u 0 on the interval l 0 or 0 for some nonzero function c c a for the neumann problem we use entirely different techniques to prove that s is never critical for any cubic polynomial j this implies at once that the neumann problem can have at most one nonconstant solution having a given number of maxima or minima this solution is necessarily strongly nondegenerate in the sense that zero is not contained in the spectrum of the linearized operator see 2 for the dirichlet problem the situation is far more complicated and the bifurcation diagrams undergo qualitative changes depending on the positions of the roots off for example fig 1 shows the bifurcation diagram for the positive solutions for two different cubic functions of the form f u a u u b u c it is interesting to note however that in every case we study there are at most three solutions to the dirichlet problem for each

generic multiplicity for a scalar field equation on compact surfaces we prove generic multiplicity of solutions for a scalar field equation on compact surfaces via morse inequalities in particular our result improves significantly the multiplicity estimate which can be deduced from the degree counting formula in chen and lin 2003 12 related results are derived for the prescribed q curvature equation

the derivation of chemotaxis equations as limit dynamics of moderately interacting stochastic many particle systems the chemotaxis equations are a well known system of partial differential equations describing aggregation phenomena in biology in this paper they are rigorously derived from an interacting stochastic many particle system where the interaction between the particles is rescaled in a moderate way as population size tends to infinity the novelty of this result is that in all previous applications of this kind of limiting procedure the principal part of the system is assumed to fulfill an ellipticity condition which is not satisfied in our case new techniques which deal with this difficulty are presented

a micropolar theory of finite deformation and finite rotation multiplicative elastoplasticity the aim of this work is to formulate a geometrically exact theory of finite deformation and finite rotation micropolar elastoplasticity to obtain a generalized nonlinear continuum framework to this end the classical deformation map is supplemented by an independent rotation field to yield an enhanced configuration space thereby the rotational part of the formulation is consequently parameterized in terms of the rotation pseudo vector via the euler rodrigues formula then micropolar hyperelasticity and multiplicative elastoplasticity are conceptionally derived as in the classical boltzmann continuum the proposed theory is consequently developed in a modern geometry oriented fashion linearization of the kinematics retrofits the well known structure of the micropolar geometrically linear theory

the mixmaster universe model revisited we re examine the singularity structure of the mixmaster universe model based on recent developments concerning the treatment of negative resonances in a previous publication we have shown that this model satisfied the ablowitz ramani segur test for the painleve property now using a different method called the perturbative painleve approach which involves series in both positive and negative powers of the independent variable we find that some compatibilities related to negative resonances are not satisfied this leaves the possibility open that the mixmaster model may be non integrable

on the integrability of bianchi cosmological models in this paper we investigate the problem of the integrability of bianchi class a cosmological models this class of systems is reduced to the form of hamiltonian systems with exponential potential forms the dynamics of bianchi class a models is investigated through the euler lagrange equations and geodesic equations in the jacobi metric on this basis we have come to some general conclusions concerning the evolution of the volume of 3 space of constant time the general form of this function has been found it can serve as a controller during numerical calculations of the dynamics of cosmological models the integrability of cosmological models is also discussed from the points of view of different integrability criteria we show that the dimension of the phase space of bianchi class a hamiltonian systems can be reduced by two we prove that the vector field of the reduced system is polynomial and that it does not admit any analytic or even formal first integral

stability of solutions of chemotaxis equations in reinforced random walks in this paper we consider a nonlinear system of differential equations consisting of one parabolic equation and one ordinary differential equation the system arises in chemotaxis a process whereby living organisms respond to chemical substance by moving toward higher or lower concentrations of the chemical substance or by aggregating or dispersing we prove that stationary solutions of the system are asymptotically stable

asymptotic high order mass preserving schemes for a hyperbolic model of chemotaxis we introduce a new class of finite difference schemes for approximating the solutions to an initial boundary value problem on a bounded interval for a one dimensional dissipative hyperbolic system with an external source term which arises as a simple model of chemotaxis since the solutions to this problem may converge to nonconstant asymptotic states for large times standard schemes usually fail to yield a good approximation therefore we propose a new class of schemes which use an asymptotic higher order correction second and third order in our examples to balance the effects of the source term and the influence of the asymptotic solutions special care is needed to deal with boundary conditions to avoid harmful loss of mass convergence results are proved for these new schemes and several numerical tests are presented and discussed to verify the effectiveness of their behavior

bubble towers for supercritical semilinear elliptic equations we construct positive solutions of the semilinear elliptic problem u u up 0 with dirichet boundary conditions in a bounded smooth domain rn n 4 when the exponent p is supercritical and close enough to n 2n 2 and the parameter r is small enough as p n 2n 2 the solutions have multiple blow up at finitely many points which are the critical points of a function whose definition involves green s function our result extends the result of del pino et al j differential equations 193 2 2003 280 when is a ball and the solutions are radially symmetric

asymptotic convergence of solutions of keller segel equations in network shaped domains abstract the objective of this study is to examine the classical keller segel equations in network shaped domains first the existence of unique strict global solutions of the equations is demonstrated by using the theory of strongly elliptic differential operators in network shaped domains and the theory of abstract parabolic semilinear evolution equations furthermore by considering an angle condition as well as the ojasiewicz simon inequality it is proved that every global solution converges to a stationary solution and the convergence is dominated by the values of a lyapunov function along the global solution

a criterion for the non existence of an additional integral in hamiltonian systems with a homogeneous potential abstract a non integrability criterion on the basis of ziglin s theorem is given for two degrees of freedom hamiltonians with a homogeneous potential of integer degree as simple examples the anisotropic kepler problem and the one dimensional newtonian three body problem are proved to be non integrable a self contained proof of ziglin s theorem n dimensional homogeneous potential version is also given

spatio temporal chaos in a chemotaxis model abstract in this paper we explore the dynamics of a one dimensional keller segel type model for chemotaxis incorporating a logistic cell growth term we demonstrate the capacity of the model to self organise into multiple cellular aggregations which according to position in parameter space either form a stationary pattern or undergo a sustained spatio temporal sequence of merging two aggregations coalesce and emerging a new aggregation appears this spatio temporal patterning can be further subdivided into either a time periodic or time irregular fashion numerical explorations into the latter indicate a positive lyapunov exponent sensitive dependence to initial conditions together with a rich bifurcation structure in particular we find stationary patterns that bifurcate onto a path of periodic patterns which prior to the onset of spatio temporal irregularity undergo a periodic doubling sequence based on these results and comparisons with other systems we argue that the spatio temporal irregularity observed here describes a form of spatio temporal chaos we discuss briefly our results in the context of previous applications of chemotaxis models including tumour invasion embryonic development and ecology

long time behaviour of solutions to a chemotaxis model with volume filling effect the lyapunov functional is constructed for a quasilinear parabolic system which models chemotaxis and takes into account a volume filling effect for some typical case it is proved that the limit set of any trajectory consists of regular stationary solutions some lower and upper bounds on the stationary solutions are found for a given range of parameters there are stationary solutions which are inhomogeneous in space

integrability of the bianchi ix system in this paper we apply the darboux theory of integrability to the classical bianchi ix system thus we provide a complete description of the darboux polynomials exponential factors rational first integrals and darboux first integrals for the classical bianchi ix system

from periodic ode s to supercritical pde s abstract let m g be an m dimensional compact riemannian manifold without boundary assume c 2 m is such that g is coercive we prove the existence of solutions to the supercritical problems g u u u p u 0 in m g and g u u e u in m g which concentrate along an m 1 dimensional submanifold of m as p and 0 respectively under suitable symmetry assumptions on the manifold m a connection with the solutions of a class of periodic ode s is established

nodal solutions for an elliptic problem involving large nonlinearities abstract let us consider the problem 0 1 u a x u u p 1 u in b 1 u 0 on b 1 where b 1 is the unit ball in r n n 3 and a x 0 is a smooth radial function under some suitable assumptions on the regular part of the green function of the operator u n 1 r u a r u we prove the existence of a changing sign solution to 0 1 for p large enough

stability of constant steady states and existence of unbounded solutions in time to a reaction diffusion equation modelling chemotaxis abstract we consider the initial boundary value problem to a reaction diffusion equation modelling chemotaxis in one space dimension under neumann boundary conditions we study the stability of constant steady states and show that there exists an unbounded solution in time induced by the instability of constant steady states

concentrating solutions for a planar elliptic problem involving nonlinearities with large exponent abstract we consider the boundary value problem u u p 0 in a bounded smooth domain in r 2 with homogeneous dirichlet boundary condition and p a large exponent we find topological conditions on which ensure the existence of a positive solution u p concentrating at exactly m points as p in particular for a nonsimply connected domain such a solution exists for any given m 1

global classical solvability and generic infinite time blow up in quasilinear keller segel systems with bounded sensitivities abstract the chemotaxis system u t d u v u s u v v v t v v u is considered under homogeneous neumann boundary conditions in a bounded domain r n n 2 along with initial conditions involving suitably regular and nonnegative data it is firstly asserted that if the positive smooth function d decays at most algebraically with respect to u then for any smooth nonnegative and bounded s fulfilling a further mild assumption especially satisfied when s s u with s 0 0 possesses a globally defined classical solution if is a ball then under appropriate assumptions on d and s generalizing the prototypical choices in d u v u 1 m 1 and s u v u u 1 1 u 0 v 0 with m r and r such that m n 2 n 0 the phenomenon of infinite time blow up is next shown to occur for all initial data within a set b of functions which inter alia is found to be dense in the set of all radially symmetric and suitably regular positive functions on up to equality in thereby covering the largest possible range of nonpositive for the appearance of unbounded solutions this extends previous findings on blow up in infinite time which in the context of were limited to a smaller parameter region and which were restricted to mere existence results without information on the richness of b

blow up analysis for su 3 toda system abstract using the method of symmetrization and the rescaling we study non compact solution sequence to the su 3 toda system in non abelian relativistic self dual gauge theory i e the quantization of the total mass and classification of the singular limit

on a conjecture of lin ni for a semilinear neumann problem let q be a bounded domain in rn n 3 and a 0 we consider au au u n 2 n 2 in q2 u 0 in q au av 0 on i2 and show that for a sufficiently small the minimal energy solutions are only constants

homoclinic heteroclinic and periodic orbits for a class of indefinite hamiltonian systems si e est l espace euclidien de produit interieur 1 on etudie les orbites d une classe de systemes hamiltoniens dans e e quand la fonctionnelle d energie est donnee par h q p 1 2 sp p v q q p e e et la forme quadratique sp p est indefinie mais non degeneree