the mckean vlasov equation in finite volume we study the mckean vlasov equation on the finite tori of length scale l in d dimensions we derive the necessary and sufficient conditions for the existence of a phase transition which are based on the criteria first uncovered in gates and penrose commun math phys 17 194 209 1970 and kirkwood and monroe j chem phys 9 514 526 1941 therein and in subsequent works one finds indications pointing to critical transitions at a particular model dependent value of the interaction parameter we show that the uniform density which may be interpreted as the liquid phase is dynamically stable for and prove abstractly that a critical transition must occur at however for this system we show that under generic conditions l large d 2 and isotropic interactions the phase transition is in fact discontinuous and occurs at some theta text t theta sharp finally for h stable bounded interactions with discontinuous transitions we show that with suitable scaling the theta text t l tend to a definitive non trivial limit as l

asymptotic expansion of solutions to the drift diffusion equation with large initial data abstract we consider the large time behavior of the solution to the initial value problem for the nernst planck type drift diffusion equation in whole spaces in the l p framework the global existence and the decay of the solution were shown moreover the second order asymptotic expansion of the solution as t was derived we also deduce the higher order asymptotic expansion of the solution especially we discuss the contrast between the odd dimensional case and the even dimensional case

asymptotics of a fredholm determinant corresponding to the first bulk critical universality class in random matrix models we study the determinant det i kpii of an integrable fredholm operator kpii acting on the interval s s whose kernel is constructed out of the function associated with the hastings mcleod solution of the second painleve equation this fredholm determinant describes the critical behavior of the eigenvalue gap probabilities of a random hermitian matrix chosen from the unitary ensemble in the bulk double scaling limit near a quadratic zero of the limiting mean eigenvalue density using the riemann hilbert method we evaluate the large s asymptotics of det i kpii

uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory we consider asymptotics for orthogonal polynomials with respect to varying exponential weights wn x dx e nv x dx on the line as n the potentials v are assumed to be real analytic with sufficient growth at infinity the principle results concern plancherel rotach type asymptotics for the orthogonal polynomials down to the axis using these asymptotics we then prove universality for a variety of statistical quantities arising in the theory of random matrix models some of which have been considered recently in 31 and also in 4 an additional application concerns the asymptotics of the recurrence coefficients and leading coefficients for the orthonormal polynomials see also 4 the orthogonal polynomial problem is formulated as a riemann hilbert problem following 19 20 the riemann hilbert problem is analyzed in turn using the steepest descent method introduced in 12 and further developed in 11 13 a critical role in our method is played by the equilibrium measure d v for v as analyzed in 8 1999 john wiley sons inc

chemotaxis with logistic source very weak global solutions and their boundedness properties abstract we consider the chemotaxis system u t u u v g u x t 0 0 v v u x t 0 in a smooth bounded domain r n where 0 and g generalizes the logistic function g u a u b u with 1 a 0 and b 0 a concept of very weak solutions is introduced and global existence of such solutions for any nonnegative initial data u 0 l 1 is proved under the assumption that 2 1 n moreover boundedness properties of the constructed solutions are studied inter alia it is shown that if b is sufficiently large and u 0 l has small norm in l for some n 2 then the solution is globally bounded finally in the case that additionally n 2 holds a bounded set in l can be found which eventually attracts very weak solutions emanating from arbitrary l 1 initial data the paper closes with numerical experiments that illustrate some of the theoretically established results

on the existence of local strong solutions to chemotaxis shallow water system with large data and vacuum abstract in this paper motivated by the chemotaxis navier stokes system arising from mathematical biology 43 a modified shallow water type chemotactic model is derived for large initial data allowing vacuum the local existence of strong solutions together with the blow up criterion is established

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

reinforced random walk leta i i 1 be a sequence of nonnegative numbers difine a nearest neighbor random motion overrightarrow x x 0 x 1 on the integers as follows initially the weight of each interval i i 1 i an integer equals 1 if at timen an interval i i 1 has been crossed exactlyk times by the motion its weight is 1 sum limits j 1 k a j given x 0 x 1 x n i0 i1 in the probability thatx n 1 isi n 1 ori n 1 is proportional to the weights at timen of the intervals i n 1 i n and i n iin 1 we prove that overrightarrow x either visits all integers infinitely often a s or visits a finite number of integers eventually oscillating between two adjacent integers a s and that mathop lim limits n to infty x n n 0 a s for much more general reinforcement schemes we provep overrightarrow x visits all integers infinitely often p overrightarrow x has finite range 1

nonlinear aggregation diffusion equations radial symmetry and long time asymptotics we analyze under which conditions equilibration between two competing effects repulsion modeled by nonlinear diffusion and attraction modeled by nonlocal interaction occurs this balance leads to continuous compactly supported radially decreasing equilibrium configurations for all masses all stationary states with suitable regularity are shown to be radially symmetric by means of continuous steiner symmetrization techniques calculus of variations tools allow us to show the existence of global minimizers among these equilibria finally in the particular case of newtonian interaction in two dimensions they lead to uniqueness of equilibria for any given mass up to translation and to the convergence of solutions of the associated nonlinear aggregation diffusion equations towards this unique equilibrium profile up to translations as t rightarrow infty

global boundedness in a fully parabolic quasilinear chemotaxis system with singular sensitivity abstract in this paper we study the global boundedness of solutions to the quasilinear fully parabolic chemotaxis system u t d u u s u v v t v v u where bounded domain r n n 2 subject to the non flux boundary conditions the diffusivity fulfills d u a 0 u 1 with a 0 0 and 0 while the density signal governed sensitivity satisfies 0 s u b 0 u 1 and 0 v v k for b 0 0 and k r it is shown that the solution is globally bounded provided 1 and k 1 this result demonstrates the effect of signal dependent sensitivity on the blow up prevention

factorization of minty and stampacchia variational inequality systems abstract in this paper sufficient regularity and coercivity conditions for minty and stampacchia type variational inequality systems are offered the typical results state that if the independent inequalities are solvable and the functions involved are lower semicontinuous then the system has also a solution that is a factorization principle holds as an application a variational inequality system consisting of two partial differential inequalities is considered this result is analogous to that of chen journal of mathematical analysis and application 231 1999 177 obtained for one variational inequality

energy type estimates and global solvability in a two dimensional chemotaxis haptotaxis model with remodeling of non diffusible attractant this paper deals with the coupled chemotaxis haptotaxis model ut u u v u w u 1 u w x t 0 0 v u v x t 0 wt vw w 1 w u x t 0 which was initially proposed by chaplain and lolas 2006 10 to describe the interactions between cancer cells the matrix degrading enzyme and the host tissue in a process of cancer cell invasion of tissue extracellular matrix here r2 is a bounded domain with smooth boundary and and are positive parameters as compared to previous mathematical studies the novelty here consists of allowing for positive values of reflecting processes of self remodeling of the extracellular matrix under zero flux boundary conditions it is shown that for any sufficiently smooth initial data u0 w0 satisfying first order compatibility conditions the model admits a unique global smooth solution a crucial ingredient in the proof is an energy like inequality which given t 0 yields boundedness of u t in llogl this serves as a starting point for a bootstrap argument used to derive higher regularity estimates sufficient for global extensibility of solutions

singularity formation in chemotaxis a conjecture of nagai consider the initial boundary value problem for the system s ut uxx uvx x vt u av on an interval 0 1 for t 0 where a 0 with ux 0 t ux 1 t 0 suppose mu v0 are positive constants the corresponding spatially homogeneous global solution u t mu v t mu a v0 mu a exp at is stable in the sense that if mu v0 are positive constants the corresponding spatially homogeneous solution will be uniformly close to u cdot v cdot we consider in sequence space an approximate system s which is related to s in the following sense the chemotactic term uvx x is replaced by the inverse fourier transform of the finite part of the convolution integral for the fourier transform of uvx x here the finite part of the convolution on the line at a point x of two functions f g is defined as int 0 x f y g y x dy we prove the following if mu a then in every neighborhood of mu v0 there are spatially nonconstant initial data for which the solution of proble

on a singular perturbation problem arising in the theory of evolutionary distributions evolution by natural selection is a process by which progeny inherit some properties from their progenitors with small variation these properties are subject to natural selection and are called adaptive traits and carriers of the latter are called phenotypes the distribution of the density of phenotypes in a population is called evolutionary distributions ed we analyze mathematical models of the dynamics of a system of ed such systems are anisotropic in that diffusion of phenotypes in each population equation remains positive in the directions of their own adaptive space and vanishes in the directions of the other s adaptive space we prove that solutions to such systems exist in a sense weaker than the usual we develop an algorithm for numerical solutions of such systems finally we conduct numerical experiments with a model in which populations compete that allow us to observe salient attributes of a specific system of ed

on pseudomonotone set valued mappings abstract in this paper a common generalization of the algebraic and topological pseudomonotonicity for set valued maps is introduced existence results for variational inequalities governed by such maps are given

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

riemann hilbert approach to a generalised sine kernel and applications we investigate the asymptotic behavior of a generalized sine kernel acting on a finite size interval q q we determine its asymptotic resolvent as well as the first terms in the asymptotic expansion of its fredholm determinant further we apply our results to build the resolvent of truncated wiener hopf operators generated by holomorphic symbols finally the leading asymptotics of the fredholm determinant allows us to establish the asymptotic estimates of certain oscillatory multidimensional coupled integrals that appear in the study of correlation functions of quantum integrable models

largest eigenvalue distribution in the double scaling limit of matrix models a coulomb fluid approach using thermodynamic arguments we find that the probability that there are no eigenvalues in the interval s infinity in the double scaling limit of hermitian matrix models is o exp s2 gamma 2 as s to infinity here gamma m 1 2 m 1 2 determines the mth multi critical point of the level density sigma x alpha b 1 x b 2 gamma x in b b and b2 alpha n furthermore the size of the transition zone where the eigenvalue density becomes vanishingly small at the tail of the spectrum is approximately n gamma 1 2 gamma 1 in agreement with earlier work based upon the string equation

existence of global bounded classical solution to a quasilinear attraction repulsion chemotaxis system with logistic source abstract this paper concerns the global existence of bounded classical solution to a quasilinear attraction repulsion chemotaxis system with logistic source u t d u u u v u w u u 2 in v t v u v in w t w u w in under homogeneous neumann boundary condition with positive parameters and d s c 0 s m 1 for s 0 d 0 0 we prove that in dimension three there exists a unique global bounded classical solution provided that m 6 5 and 0 with an additional assumption d s c 0 s m 1 1 for s 0 we prove that for any m 1 6 5 there exists a constant 0 0 m such that for all 0 the above problem admits a unique global bounded classical solution

analysis for a system of coupled reaction diffusion parabolic equations arising in biology we study a system where the equations are coupled by transport and reaction terms in the general case we prove local existence and uniqueness of the solution and in a particular case for a priori fixed time t global existence on 0 t

global existence versus blow up in a high dimensional keller segel equation with degenerate diffusion and nonlocal aggregation abstract in this paper we study the degenerate keller segel equation with nonlocal aggregation u t u m u b u in r d r where m 1 d 3 and b u 2 u with 2 d by analyzing the interaction between the degenerate diffusion and the nonlocal aggregation we determine the conditions for initial data under which weak solutions globally exist or blow up in finite time with m 1 d d d furthermore a sharper criterion is given for global existence and finite time blow up of weak solutions with m in the subrange 2 d 2 d d d 1 d d

on the semi monotone operator theory and applications abstract in this paper eis a real banach space a e e e is a semi monotone operator we first study the variational inequality problem findu k such that a u u v u 0 wherek e is a closed convex subset then the operator equationa u u p and we also construct a degree theory for demi continuous semi monotone operators in reflexive banach spaces

dynamics in a parabolic elliptic two species population competition model with cross diffusion for one species abstract this work deals with a simplified parabolic elliptic population competition model with cross diffusion for one species we first claim that this model has at least a non trivial global smooth solution and then we detect explicit parameter ranges under which the coexistence steady state and the two semi trivial steady states are globally and asymptotically stable

on a nonlocal phase separation model abstract an alternative to the cahn hilliard model of phase separation for two phase systems in a simplified isothermal case is given the model is derived from a free energy with a nonlocal interacting term and allows reasonable bounds for the concentrations using the free energy as lyapunov functional the asymptotic state of the system is investigated and characterized by a variational principle

many body theory of chemotactic cell cell interactions we consider an individual based stochastic model of cell movement mediated by chemical signaling fields this model is formulated using langevin dynamics which allows an analytic study using methods from statistical and many body physics in particular we construct a diagrammatic framework within which to study cell cell interactions in the mean field limit where statistical correlations between cells are neglected we recover the deterministic keller segel equations within exact perturbation theory in the chemotactic coupling epsilon statistical correlations are non negligible at large times and lead to a renormalization of the cell diffusion coefficient d r an effect that is absent at mean field level an alternative closure scheme based on the necklace approximation probes the strong coupling behavior of the system and predicts that d r is renormalized to zero at a critical value of epsilon indicating self localization of the cell stochastic simulations of the model give very satisfactory agreement with the perturbative result at higher values of the coupling simulations indicate that d r approximately epsilon 2 a result at odds with the necklace approximation we briefly discuss an extension of our model which incorporates the effects of short range interactions such as cell cell adhesion

application of an adams type inequality to a two chemical substances chemotaxis system abstract this paper deals with positive solutions of the fully parabolic system u t u u v in 0 1 v t v v w in 0 2 w t w w u in 0 under homogeneous neumann boundary conditions or mixed boundary conditions no flux and dirichlet conditions in a smooth bounded domain r n n 4 with positive parameters 1 2 0 and nonnegative smooth initial data u 0 v 0 w 0 in the lower dimensional case n 3 it is proved that for all reasonable initial data solutions of the system exist globally in time and remain bounded in the case n 4 it is shown that in the radially symmetric setting solutions to the neumann boundary value problem of the system exist globally in time and remain bounded if u 0 l 1 8 2 as to the mixed boundary value problem we will establish global existence and boundedness of solutions if u 0 l 1 8 2 without radial symmetry the key ingredients are a lyapunov functional and an adams type inequality a lyapunov functional of the above problems will be constructed and the constant 8 2 is deduced from the critical constant in the adams type inequality this result is regarded as a generalization of the well known 8 problem in the keller segel system to higher dimensions

on explosions of solutions to a system of partial differential equations modelling chemotaxis a system of partial differential equations modelling chemotactic aggregation is analysed keller segel model conditions on the system of paramaters are given implying global existence of smooth solutions in two space dimensions and radially symmetric situations explosion of the bacteria concentration in finite time is shown for a class of initial values

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

blow up and decay criteria for a model of chemotaxis for a system of equations introduced by jager and luckhaus 1992 6 as a model of chemotaxis the questions of blow up and global existence criteria are investigated specifically for a convex region a lower bound for the blow up time is derived if the solution blows up and explicit criteria to ensure non blow up are determined

universality of the double scaling limit in random matrix models we study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support we establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit the limiting kernels are constructed out of functions associated with the second painleve equation this extends a result of bleher and its for the special case of a critical quartic potential the two main tools we use are equilibrium measures and riemann hilbert problems in our treatment of equilibrium measures we allow a negative density near the critical point which enables us to treat all cases simultaneously the asymptotic analysis of the riemann hilbert problem is done with the deift zhou steepest descent analysis for the construction of a local parametrix at the critical point we introduce a modification of the approach of baik deift and johansson so that we are able to satisfy the required jump properties exactly c 2005 wiley periodicals inc

generalized vector complementarity type problems in topological vector spaces in this paper we introduced the generalized vector variational inequality type problem and the generalized vector complementarity type problem in the setting of topological vector space by utilizing a modified version of the fan kkm theorem we investigated the nonemptyness and compactness of solution sets of these problems without the demipseudomonotonicity assumption further we prove that solution sets of both the problems are equivalent to each other under some suitable conditions the results of this paper generalize and improve several results that appeared recently in the literature

on the derivation of macroscopic hyperbolic equations for binary multicellular growing mixtures this paper presents an asymptotic theory based on the hyperbolic scaling for a large class of boltzmann type equations suitable to model the evolution of a binary mixture of multicellular systems in biology the modelling approach is that of the kinetic theory for active particles time scaling related to cell movement and biological activity are shown to play a crucial role in determining the macroscopic equations corresponding to each case applications are focused on the immune competition

asymptotics of the airy kernel determinant the authors use riemann hilbert methods to compute the constant that arises in the asymptotic behavior of the airy kernel determinant of random matrix theory

fixed point theory and nonlinear problems introduction among the most original and far reaching of the contributions made by henri poincare to mathematics was his introduction of the use of topological or qualitative methods in the study of nonlinear problems in analysis his starting point was the study of the differential equations of celestial mechanics and in particular of their periodic solutions his work on this topic began with his thesis in 1879 and was developed in detail in his great three volume work methodes nouvelles de la mechanique celeste which appeared in the early 1890s and summarized his many memoirs of the intervening period it continued until his memoir shortly before his death in 1912 in which he put forward the unproved fixed point result usually referred to as poincare s last geometric theorem the ideas introduced by poincare include the use of fixed point theorems the continuation method and the general concept of global analysis the writer s acquaintance with poincare s influence came through contact with solomon lefschetz and marston morse both of whom were very explicit as to the role of poincare as an initiator in this direction of mathematical development in 1934 in the foreword to his colloquium volume on the calculus of variations in the large morse had put this forward very forcefully in the first paragraph

from a discrete model of chemotaxis with volume filling to a generalized patlak keller segel model we present a discrete model of chemotaxis whereby cells responding to a chemoattractant are seen as individual agents whose movement is described through a set of rules that result in a biased rand

repulsion effects on boundedness in the higher dimensional fully parabolic attraction repulsion chemotaxis system abstract this paper deals with an attraction repulsion chemotaxis system u t d u u u v u w x t 0 1 v t v u v x t 0 2 w t w u w x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 2 where parameters i i 1 2 and are positive and diffusion coefficient d u c 2 0 satisfies d u 0 for u 0 d u d u m 1 with d 0 and m 1 for all u 0 it is proved that the corresponding initial boundary value problem possesses a unique global bounded classical solution for m 2 2 n in particular in the case 1 2 and the solution is globally bounded if m 2 2 n n 2 n 2 n 2 therefore due to the inhibition of repulsion to the attraction the range of m 2 2 n of boundedness is enlarged and the results of 21 is thus extended to the higher dimensional attraction repulsion chemotaxis system with nonlinear diffusion

dyson s constant in the asymptotics of the fredholm determinant of the sine kernel we prove that the asymptotics of the fredholm determinant of i k where k is the integral operator with the sine kernel open image in new window on the interval 0 are given by

global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening abstract this paper deals with nonnegative solutions of the neumann initial boundary value problem for the parabolic chemotaxis system u t u u v u u 2 x t 0 v t v v u x t 0 in bounded convex domains r n n 1 with smooth boundary it is shown that if the ratio is sufficiently large then the unique nontrivial spatially homogeneous equilibrium given by u v 1 is globally asymptotically stable in the sense that for any choice of suitably regular nonnegative initial data u 0 v 0 such that u 0 0 the above problem possesses a uniquely determined global classical solution u v with u v t 0 u 0 v 0 which satisfies u t 1 l 0 and v t 1 l 0 as t

the widom dyson constant for the gap probability in random matrix theory in the bulk scaling limit for the gaussian unitary ensemble in random matrix theory the probability that there are no eigenvalues in the interval 0 2s is given by p s det i k s where k s is the trace class operator with kernel k s x y sin x y p x y acting on l 2 0 2s in the analysis of the asymptotic behavior of p s as s there is particular interest in the constant term known as the widom dyson constant we present a new derivation of this constant which can be adapted to calculate similar critical constants in other problems arising in random matrix theory

asymptotics of tracy widom distributions and the total integral of a painleve ii function the tracy widom distribution functions involve integrals of a painleve ii function starting from positive infinity in this paper we express the tracy widom distribution functions in terms of integrals starting from minus infinity there are two consequences of these new representations the first is the evaluation of the total integral of the hastings mcleod solution of the painleve ii equation the second is the evaluation of the constant term of the asymptotic expansions of the tracy widom distribution functions as the distribution parameter approaches minus infinity for the gue tracy widom distribution function this gives an alternative proof of the recent work of deift its and krasovsky the constant terms for the goe and gse tracy widom distribution functions are new

asymptotic stability of a two species chemotaxis system with non diffusive chemoattractant abstract we study the behavior of two biological populations u and v attracted by the same chemical substance whose behavior is described in terms of second order parabolic equations the model considers a logistic growth of the species and the interactions between them are relegated to the chemoattractant production the system is completed with a third equation modeling the evolution of chemical we assume that the chemical w is a non diffusive substance and satisfies an ode more precisely u t u u 1 w w 1 u 1 u x t 0 v t v v 2 w w 2 v 1 v x t 0 w t h u v w x t 0 under appropriate boundary and initial conditions in an n dimensional open and bounded domain we consider the cases of positive chemo sensitivities not necessarily constant elements the chemical production function h increases as the concentration of the species u and v increases we first study the global existence and uniform boundedness of the solutions by using an iterative approach the asymptotic stability of the homogeneous steady state is a consequence of the growth of h i and the size of i finally some examples of the theoretical results are presented for particular functions h and i

on the generalized monotonicity of variational inequalities in this paper three new classes of generalized monotone operators are introduced the relaxed m pseudomonotone operator relaxed m quasimonotone operator and densely relaxed pseudomonotone operator the relations between these three classes and the relations between them and some other well known classes of generalized monotone operators are discussed in detail various existence results for variational inequalities in normed spaces are derived the results generalize some known results

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

generalized mixed equilibrium problems with generalized α η monotone bifunction in topological vector spaces the purpose of this paper is to introduce a new class of the generalized mixed equilibrium problems with a new definition of the relaxed monotonicity for bi functions in topological vector spaces by employing the kkm technique and under some appropriate assumptions on the considering nonlinear mappings we obtain the existence of a solution for the generalized mixed equilibrium problems with the new concept of the relaxed monotonicity and coercivity condition in order to relax the compactness of the domains of the nonlinear mappings in the setting of topological vector spaces moreover the compactness and convexness of the solution set are investigated the results in the paper extend and generalize the corresponding results especially sintunavarat 2013 20 in this area by providing mild assumptions in order to guarantee the existence of a solution for the generalized mixed equilibrium problem

a generalized vector variational inequality problem with a set valued semi monotone mapping abstract in this paper we first generalize the concept of semi monotonicity for a scalar single valued mapping to the case of a vector set valued mapping then we consider a generalized vector variational inequality problem concerning the vector set valued semi monotone mapping several existence results are obtained by using the kakutani fan glicksberg fixed point theorem which generalize some results for vector variational inequality problems studied in recent years

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

the basin of attraction of the steady states for a chemotaxis model in mathbb r 2 with critical mass we consider the cauchy problem for a parabolic elliptic system in mathbb r 2 which is amathematical model of chemotaxis and also amodel of self attracting particles in the critical mass case we determine the basin of attraction of the steady states for the cauchy problem through a lyapunov functional

double scaling limit in the random matrix model the riemann hilbert approach we prove the existence of the double scaling limit in the unitary matrix model with quartic interaction and we show that the correlation functions in the double scaling limit are expressed in terms of the integrable kernel determined by the function for the hastings mcleod solution to the painleve ii equation the proof is based on the riemann hilbert approach and the central point of the proof is an analysis of analytic semiclassical asymptotics for the function at the critical point in the presence of four coalescing turning points 2003 wiley periodicals inc

the diffusion limit of transport equations derived from velocity jump processes in this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a poisson process we show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion in general the resulting diffusion tensor is anisotropic and we give necessary and sufficient conditions under which it is isotropic we also give a method to construct approximations of arbitrary high order for large times in a second paper part ii we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion depending on the strength of the bias it may lead to an anisotropic diffusion equation to a drift term in the flux or to both our analysis generalizes and simplifies previous derivations that lead to the classical patlak keller segel alt

unboundedness for solutions to a degenerate drift diffusion equation with the l1 supercritical and the energy subcritical exponent abstract we consider large time behavior of weak solutions to a degenerate drift diffusion system related to keller segel system with the l 1 supercritical and the energy subcritical cases under relaxed weight condition it is known that the large time behavior of solutions is classified by the invariant norms of initial data for the l 1 critical case ogawa wakui proved that the negative entropy condition induces the unboundedness of corresponding solutions with the initial data decaying slowly at spacial infinity here the result is a continuous analogy of the l 1 critical case analogous results have been obtained in the theory of nonlinear schrodinger equations

the diffusion limit of transport equations ii chemotaxis equations in this paper we use the diffusion limit expansion of transport equations developed earlier t hillen and h g othmer siam j appl math 61 2000 pp 751 775 to study the limiting equation under a variety of external biases imposed on the motion when applied to chemotaxis or chemokinesis these biases produce modification of the turning rate the movement speed or the preferred direction of movement depending on the strength of the bias it leads to anisotropic diffusion to a drift term in the flux or to both in the parabolic limit we show that the classical chemotaxis equation which we call the patlak keller segel alt pksa equation arises only when the bias is sufficiently small using this general framework we derive phenomenological models for chemotaxis of flagellated bacteria of slime molds and of myxobacteria we also show that certain results derived earlier for one dimensional motion can easily be generalized to two or three dimensional motion as well

on a parabolic elliptic system with gradient dependent chemotactic coefficient abstract we consider a second order pdes system of parabolic elliptic type with chemotactic terms the system describes the evolution of a biological species u moving towards a higher concentration of a chemical stimuli v in a bounded and open domain of r n in the system considered the chemotaxis sensitivity depends on the gradient of v i e the chemotaxis term has the following expression d i v u v p 2 v where is a positive constant and p satisfies p 1 if n 1 and p 1 n n 1 if n 2 we obtain uniform bounds in l and the existence of global in time solutions for the one dimensional case we prove the existence of infinitely many non constant steady states for p 1 2 for any positive and a given positive mass

existence of solutions of the hyperbolic keller segel model we are concerned with the hyperbolic keller segel model with quorum sensing a model describing the collective cell movement due to chemical signalling with a flux limitation for high cell densities this is a first order quasilinear equation its flux depends on space and time via the solution to an elliptic pde in which the right hand side is the solution to the hyperbolic equation this model lacks strong compactness or contraction properties our purpose is to prove the existence of an entropy solution obtained as usual in passing to the limit in a sequence of solutions to the parabolic approximation the method consists in the derivation of a kinetic formulation for the weak limit the specific structure of the limiting kinetic equation allows for a rigidity theorem which identifies some property of the solution which might be nonunique to this kinetic equation this is enough to deduce a posteriori the strong convergence of a subsequence

infinite time aggregation for the critical patlak keller segel model in ℝ2 we analyze the two dimensional parabolic elliptic patlak keller segel model in the whole euclidean space r2 under the hypotheses of integrable initial data with finite second moment and entropy we first show local in time existence for any mass of free energy solutions namely weak solutions with some free energy estimates we also prove that the solution exists as long as the entropy is controlled from above the main result of the paper is to show the global existence of free energy solutions with initial data as before for the critical mass 8 actually we prove that solutions blow up as a delta dirac at the center of mass when t keeping constant their second moment at any time furthermore all moments larger than 2 blow up as t if initially bounded

level spacing of random matrices in an external source in an earlier work we had considered a gaussian ensemble of random matrices in the presence of a given external matrix source the measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials or on the coulomb gas representation are not available nevertheless the n point correlation functions are still given in terms of the determinant of a kernel known through an explicit integral representation this kernel is no longer symmetric though and is not readily accessible to standard methods in particular finding the level spacing probability is always a delicate problem in fredholm theory and we have to reconsider the problem within our model we find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state the problem is solved through coupled non linear differential equations which turn out to form a hamiltonian system as a result we find that the level spacing probability p s behaves like exp c s 8 over 3 for large spacing s this is consistent with the asymptotic behavior exp c s 2 beta 2 whenever the density of state behaves near the edge as rho lambda sim lambda beta

from microscopic to macroscopic description of multicellular systems and biological growing tissues this paper presents an asymptotic theory for a large class of boltzmann type equations suitable to model the evolution of multicellular systems in biology the mathematical approach described herein shows how various types of diffusion phenomena linear and nonlinear can be obtained in suitable asymptotic limits time scaling related to cell movement and biological activity are shown to play a crucial role in determining the macroscopic equations corresponding to each case

universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles this paper is devoted to the rigorous proof of the universality conjecture of random matrix theory according to which the limiting eigenvalue statistics ofn n random matrices within spectral intervals ofo n 1 is determined by the type of matrix real symmetric hermitian or quaternion real and by the density of states we prove this conjecture for a certain class of the hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function the potential in the quantum field theory satisfying some regularity conditions

a generalized mixed vector variational like inequality problem abstract in this paper we introduce relaxed p monotone mapping and by utilizing kkm technique and nadler s lemma we establish some existence results for the generalized mixed vector variational like inequality problem further we give the concepts of complete semicontinuity and strong semicontinuity and prove the solvability for generalized mixed vector variational like inequality problem without monotonicity assumption by applying the brouwer s fixed point theorem the results presented in this paper are extensions and improvements of some earlier and recent results in the literature

local null controllability for a chemotaxis system of parabolic elliptic type abstract we consider the controllability of a chemotaxis system of parabolic elliptic type by linearizing the nonlinear system into two separate linear equations we can bypass the obstacle caused by the nonlinear drift term and establish local null controllability of the original nonlinear system this approach is different from the usual method for dealing with coupled parabolic systems

existence results for stampacchia and minty type implicit variational inequalities with multivalued maps abstract stampacchia and minty type generalized implicit variational inequality problems are considered we extended the notion of dense pseudomonotonicity for multivalued maps and established several existence results for solutions of these problems in the setting of segment dense sets we also studied the existence of solutions of minty type generalized implicit quasi variational inequality problems some particular cases are also studied it is shown that our results contain several existing results in the literature as special cases

tracy widom distributions in critical unitary random matrix ensembles and the coupled painleve ii system we study fredholm determinants of the painleve ii and painleve xxxiv kernels in certain critical unitary random matrix ensembles these determinants describe special gap probabilities of eigenvalues we obtain tracy widom formulas for the fredholm determinants which are explicitly given in terms of integrals involving a family of distinguished solutions to the coupled painleve ii system in dimension four moreover the large gap asymptotics for these fredholm determinants are derived where the constant terms are given explicitly in terms of the riemann zeta function

global existence of solutions for a strongly coupled quasilinear parabolic system we are interested in existence uniqueness and a priori estimates of non negative solutions to the following problem begin gathered u t delta left left 1 alpha v right u right uf u v hfill v t delta v vg u v quad in 0 infty times omega hfill u v 0 quad in 0 infty times partial omega hfill u 0 u 0 v 0 v 0 quad in omega hfill end gathered 1 0

equilibrium problems with set valued mappings in banach spaces abstract in this paper we consider two classes of equilibrium problems which imply general pseudomonotonicity and general semipseudomonotonicity relative to a set valued map on reflexive and nonreflexive banach spaces respectively using the classical kkm technique we obtain some existence results under boundedness as well as under unboundedness assumptions some examples are given

asymptotics of a τ function arising in the two dimensional ising model the short distance assymptotics of the function associated to the 2 point function of the two dimensional ising model is computed as a function of the integration constant defined from the long distance behavior of the function the result is expressible in terms of the barnes double gamma function equivalently the barnesg function

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

semiclassical asymptotics of orthogonal polynomials riemann hilbert problem and universality in the matrix model we derive semiclassical asymptotics for the orthogonal polynomials pn z on the line with respect to the exponential weight exp inv z where v z is a double well quartic polynomial in the limit when n n 1 we assume that n n cri for some 0 where cr is the critical value which separates orthogonal polynomials with two cuts from the ones with one cut simultaneously we derive semiclassical asymptotics for the recursive coe cients of the orthogonal polynomials and we show that these coe cients form a cycle of period two which drifts slowly with the change of the ratio n n the proof of the semiclassical asymptotics is based on the methods of the theory of integrable systems and on the analysis of the appropriate matrix riemannhilbert problem as an application of the semiclassical asymptotics of the orthogonal polynomials we prove the universality of the local distribution of eigenvalues in the matrix model with the double well quartic interaction in the presence of two cuts contents

global solutions in a fully parabolic chemotaxis system with singular sensitivity and nonlinear signal production a fully parabolic chemotaxis system with singular sensitivity and nonlinear signal production is considered under suitable assumption on the sensitivity value and the signal production term it is shown that this system admits a global classical solution the proof is based on a series of a priori estimate techniques a fully parabolic chemotaxis system with singular sensitivity and nonlinear signal production is considered under suitable assumption on the sensitivity value and the signal production term it is shown that this system admits a global classical solution the proof is based on a series of a priori estimate techniques

monodromy preserving deformation of linear ordinary differential equations with rational coefficients iii abstract a unified treatment of monodromy and spectrum preserving deformations is presented in particular a general procedure is described to reduce the latter into the former consistently the concept of the function previously introduced for the former 2 is extended to the isospectral context it is shown that the general monodromy and spectrum preserving deformation equations can be written as hirota s bilinear differential equations by using the functions as dependent variables

higher order analogues of the tracy widom distribution and the painleve ii hierarchy we study fredholm determinants related to a family of kernels which describe the edge eigenvalue behavior in unitary random matrix models with critical edge points the kernels are natural higher order analogues of the airy kernel and

uniform convergence of proliferating particles to the fkpp equation abstract in this paper we consider a system of brownian particles with proliferation whose rate depends on the empirical measure the dependence is more local than a mean field one and has been called moderate interaction by oelschlager 16 17 we prove that the empirical process converges uniformly in the space variable to the solution of the fisher kolmogorov petrowskii piskunov equation we use a semigroup approach which is new in the framework of these systems and is inspired by some literature on stochastic partial differential equations

singularity patterns in a chemotaxis model the authors study a chemotactic model under certain assumptions and obtain the existence of a class of solutions which blow up at the center of an open disc in finite time such a finite time blow up of solutions implies chemotactic collapse namely concentration of species to form sporae the model studied is the limiting case of a basic chemotactic model when diffusion of the chemical approaches infinity which has the form ut u u v 0 v u 1 on r2 where is an open disc with no flux homogeneous neumann boundary conditions the initial conditions are continuous functions u x 0 u0 x 0 v x 0 v0 x 0 for x under these conditions the authors prove there exists a radially symmetric solution u r t which blows up at r 0 t t a specific description of such a solution is presented the authors also discuss the strong similarity between the chemotactic model they study and the classical stefan problem

from gap probabilities in random matrix theory to eigenvalue expansions we present a method to derive asymptotics of eigenvalues for trace class integral operators acting on a single interval which belongs to the ring of integrable operators its et al 1990 int j mod phys b 4 1003 37 our emphasis lies on the behavior of the spectrum of k as and i is fixed we show that this behavior is intimately linked to the analysis of the fredholm determinant as and in a stokes type scaling regime concrete asymptotic formulae are obtained for the eigenvalues of airy and bessel kernels in random matrix theory

on a parabolic elliptic system with chemotaxis and logistic type growth we consider a nonlinear pdes system of two equations of parabolic elliptic type with chemotactic terms the system models the movement of a biological population u towards a higher concentration of a chemical agent w in a bounded and regular domain rn for arbitrary n n after normalization the system is as follows ut u div um w u 1 u in t 0 t w w u in t for some positive constants m and with positive initial datum u0 and neumann boundary conditions we study the range of parameters and constrains for which the solution exists globally in time if either m 1 or m 1 and n 22 m 1 n the solution exists globally in time moreover if m 1 and 2 and there exist positive constants u 0 and u 0 such that 0 u 0 u0 u 0 we have that u 1 l w 1 l 0 as t