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

a class of p q laplacian type equation with concave convex nonlinearities in bounded domain abstract in this paper our main purpose is to establish the existence of multiple solutions of a class of p q laplacian equation involving concave convex nonlinearities p u q u v x u r 2 u u p 2 u f x u x u 0 x where is a bounded domain in r n 0 1 r q p n and p n p n p is the critical sobolev exponent s u div u s 2 u is the s laplacian of u we prove that for any 0 0 is a constant there is a 0 such that for every 0 the above problem possesses infinitely many weak solutions we also obtain some results for the case 1 q p r p the existence results of solutions are obtained by variational methods

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

a finite element method for equal width equation abstract a numerical solution of the equal width ew equation is obtained using space splitting technique and quadratic b spline galerkin finite element method solitary wave motion interaction of two solitary waves wave undulation and wave generation are studied using the proposed method comparisons are made with analytical solutions and with some spline finite element method calculations at selected times accuracy and efficiency are discussed by computing the numerical conserved laws and l 2 l error norms

a mesh free numerical method for solution of the family of kuramoto sivashinsky equations in this paper radial basis function rbfs based mesh free method is implemented to find numerical solution of the kuramoto sivashinsky equations this approach has an edge over traditional methods such as finite difference and finite element methods because it does not require a mesh to discretize the problem domain and a set of scattered nodes in the domain of influence provided by initial data is required for the realization of the method the accuracy of the method is assessed in terms of the error norms l 2 l number of nodes in the domain of influence free parameter dependent parameter rbfs and time step length numerical experiments demonstrate accuracy and robustness of the method for solving a class of nonlinear partial differential equations

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

numerical simulation of equal width wave equation by means of the variational iteration method the solution of the equal width wave equation is numerically obtained and compared with that obtained by the adomian decomposition method numerical results show that the variational iteration method can readily be implemented with excellent accuracy to nonlinear equations

constant sign and nodal solutions of coercive p q laplacian problems abstract the existence of a greatest negative a smallest positive and a nodal weak solution to a dirichlet problem with p q laplacian and reaction term that makes coercive the corresponding energy functional is investigated via variational methods besides truncation techniques

on the solutions of the p q laplacian problem at resonance abstract we consider a resonance problem driven by a nonhomogeneous operator and a caratheodory reaction f using a variant of the monotone operator theorem we show that the problem has at least a nontrivial solution

application of galerkin s method to equal width wave equation the non linear equal width equation is solved by galerkin s method using linear finite elements a linear recurrence relationship for the numerical solution of the resulting system of ordinary differential equations is found employing the crank nicolson approach including a product approximation the three invariants of the motion are calculated to determine the conservation properties of the system l 2 and l error norms are used to measure differences between the exact and numerical solutions the simulations of solitary wave motion are used to determine the properties of the algorithm finally the development of an undular bore is studied

s shaped bifurcation curves for nonlinear two parameter problems abstract we consider the nonlinear eigenvalue problem u t 1 u t u t 2 u t p u t 0 t i 1 2 1 2 u 1 2 u 1 2 0 where p 2 is a constant 0 and 0 are parameters this equation was introduced by crandall and rabinowitz 1973 as the typical model which develops the s shaped bifurcation curve when p 3 and 0 1 it is known that if p 2 and 0 1 then there exists a unique solution u r c 2 i with u where 0 is a given constant furthermore is s shaped in this paper we establish the asymptotic formulas for as 0 1 2 when 0 1 where 1 and 2 are two turning points of and is the unique positive solution to the algebraic equation 1 x x 2 x p 0 we also characterize the asymptotic shape of u t as by establishing the asymptotic behavior of u q as when 0 1 is a fixed constant

mean field treatment of the many body fokker planck equation we review some properties of the stationary states of the fokker planck equation for n interacting particles within a mean field approximation which yields a non linear integrodifferential equation for the particle density analytical results show that for attractive long range potentials the steady state is always a precipitate containing one or several clusters of small size for arbitrary potential linear stability analysis allows the statement of the conditions under which the uniform equilibrium state is unstable against small perturbations and via the einstein relation definition of a critical temperature tc separating the two phases uniform and precipitate the corresponding phase diagram turns out to be strongly dependent on the pair potential in addition numerical calculations reveal that the transition is hysteretic we finally discuss the dynamics of relaxation for the uniform state suddenly cooled below tc

exact multiplicity and bifurcation diagrams of positive solutions of a one dimensional multiparameter prescribed mean curvature problem abstract we study the exact multiplicity and bifurcation diagrams of positive solutions u c 2 l l c l l of the one dimensional multiparameter prescribed mean curvature problem u x 1 u x 2 u p u q l x l u l u l 0 where 0 is a bifurcation parameter l 0 the radius of the one dimensional ball l l is an evolution parameter and 0 p q are two constants we prove that the problem has at most two positive solutions for any 0 p q and l 0 in addition if 0 p q q p p 1 2 p 1 we give a classification of totally three qualitatively different bifurcation diagrams on the u plane for any l 0 for any fixed p 0 and q q p p 2 2 2 p 3 we prove that there exist positive l l such that the bifurcation diagrams on the u plane are individually qualitatively different for the cases i 0 l l or l l ii l l iii l l l

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

maximum likelihood estimation of mckean vlasov stochastic differential equation and its application mckean vlasov stochastic differential equation is a class of complicated and special equation since the drift term is a function of stochastic process and its distribution this paper discusses the maximum likelihood estimation of parameters in the drift term through transforming mckean vlasov stochastic process into homogeneous one and estimates parameters of the latter to discuss that of mckean vlasov equation then we build a mckean vlasov stochastic model for ion diffusion since ions moved by liquid viscous force and also by coulomb interaction related with ion charged distribution and simulate the changing trajectory of the ion motion through numerical calculation results manifest that the ion motion shows strong random property and has the same tendency for different time intervals however the smaller of time lag the more distinct of wave trajectory observed

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

global bifurcation diagrams and exact multiplicity of positive solutions for a one dimensional prescribed mean curvature problem arising in mems abstract we study global bifurcation diagrams and exact multiplicity of positive solutions for the one dimensional prescribed mean curvature problem arising in mems u x 1 u x 2 1 u p u 1 l x l u l u l 0 where 0 is a bifurcation parameter and p l 0 are two evolution parameters we determine the exact number of positive solutions by the values of p l and moreover for p 1 the bifurcation diagram undergoes fold and splitting bifurcations while for 0 p 1 the bifurcation diagram undergoes fold splitting and segment shrinking bifurcations our results extend and improve those of brubaker and pelesko n d brubaker j a pelesko analysis of a one dimensional prescribed mean curvature equation with singular nonlinearity nonlinear anal 75 2012 5086 5102 and pan and xing h pan r xing exact multiplicity results for a one dimensional prescribed mean curvature problem related to a mems model nonlinear anal rwa 13 2012 2432 2445 by generalizing the nonlinearity 1 u 2 to 1 u p with general p 1 we also answer an open question raised by brubaker and pelesko on the extension of global bifurcation diagram results to general p 0 concerning this open question we find and prove that global bifurcation diagrams for 0 p 1 are different to and more complicated than those for p 1

self organization in systems of self propelled particles we investigate a discrete model consisting of self propelled particles that obey simple interaction rules we show that this model can self organize and exhibit coherent localized solutions in one and in two dimensions in one dimension the self organized solution is a localized flock of finite extent in which the density abruptly drops to zero at the edges in two dimensions we focus on the vortex solution in which the particles rotate around a common center and show that this solution can be obtained from random initial conditions even in the absence of a confining boundary furthermore we develop a continuum version of our discrete model and demonstrate that the agreement between the discrete and the continuum model is excellent

a petrov galerkin method for solving the generalized equal width gew equation the generalized equal width gew equation is solved numerically by the petrov galerkin method using a linear hat function as the test function and a quadratic b spline function as the trial function product approximation has been used in this method a linear stability analysis of the scheme shows it to be conditionally stable test problems including the single soliton and the interaction of solitons are used to validate the suggested method which is found to be accurate and efficient finally the maxwellian initial condition pulse is studied

the classification of the single traveling wave solutions to the generalized equal width equation abstract the complete discrimination system for polynomial is applied to the generalized equal width ew equation to obtain the classifications of single traveling wave solutions in particular we construct the corresponding solutions for the concrete parameters to show that each solution in the classification can be realized

multiple angle formulas of generalized trigonometric functions with two parameters abstract generalized trigonometric functions with two parameters were introduced by drabek and manasevich to study an inhomogeneous eigenvalue problem of the p laplacian concerning these functions no multiple angle formula has been known except for the classical cases and a special case discovered by edmunds gurka and lang not to mention addition theorems in this paper we will present new multiple angle formulas which are established between two kinds of the generalized trigonometric functions and apply the formulas to generalize classical topics related to the trigonometric functions and the lemniscate function

rbf ps scheme for solving the equal width equation the equal width equation is solved by a rbf ps scheme this technique is meshless and there is no need to linearize the nonlinear terms the radial kernels are used to transform the pde into a system of odes a higher order ode solver is used to solve the ode system the method is tested for single solitary wave interaction of two and three solitary waves the development of an undular bore and the interaction of two solitary waves the results are compared with earlier work and other methods

solitary waves of the regularized long wave equation abstract a finite element solution of the regularised long wave equation based on galerkin s method using cubic splines as element shape functions is set up a linear stability analysis shows the scheme to be unconditionally stable test problems including the migration and interaction of solitary waves are used to validate the method which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evolution of a maxwellian initial pulse is then studied

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

solitary waves of the equal width wave equation a numerical solution of the equal width wave equation based on galerkin s method using cubic b spline finite elements is used to simulate the migration and interaction of solitary waves the interaction of two solitary waves is seen to cause the creation of a source for solitary waves usually these are of small magnitude but when the amplitudes of the two interacting waves are equal and opposite the source produces trains of solitary waves whose amplitudes are of the same order as those of the initiating waves the three invariants of the motion are evaluated to determine the conservation properties of the system finally the temporal evolution of a maxwellian initial pulse is studied for small ut uux uxxt 0 only positive waves are formed and the behaviour mimics that of the kdv and filw equations for larger values of d both positive and negative solitary waves are generated

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

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

collocation method using quartic b spline for the equal width ew equation the equal width ew equation is solved numerically by giving a new algorithm based on collocation method using quartic b splines at the mid knots points as element shape also we use the fourth runge kutta method for solving the system of first order ordinary differential equations instead of finite difference method our test problems including the migration and interaction of solitary waves are used to validate the algorithm which is found to be accurate and efficient the three invariants of the motion are evaluated to determine the conservation properties of the algorithm the temporal evaluation of a maxwellian initial pulse is then studied

a homotopic deformation along p of a leray schauder degree result and existence for u p 2u ƒ t u 0 u 0 u t 0 p 1 soit f 0 1 r r une fonction continue et soit o p r r definie par o p s s p 2 s pour p fixe 1 on considere le probleme de dirichlet suivant o p u f t u 0 u 0 u t 0 ou t est un nombre reel positif

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

generalized jacobian elliptic functions and their application to bifurcation problems associated with p laplacian abstract the jacobian elliptic functions are generalized and applied to bifurcation problems associated with p laplacian the values of bifurcation parameter and the corresponding solutions are represented in terms of common parameters and a complete description of the bifurcation diagram and a closed form representation of the corresponding solutions are obtained as a by product of the representation it turns out that a kind of solution is also a solution of an eigenvalue problem of p 2 laplacian

analytical and computational approaches on solitary wave solutions of the generalized equal width equation abstract in this article firstly numerical solutions of the generalized equal width gew equation have been obtained by a petrov galerkin finite element method using cubic b spline base functions as element shape functions and quadratic b spline base functions as the weight functions in order to prove the practicability and robustness of the numerical algorithm the error norms l2 l and three invariants i1 i2 and i3 are computed a linear stability analysis based on a fourier method states that the numerical scheme is unconditionally stable secondly we have proposed the modified extended tanh function method with the riccati differential equation which is a convenient and an effective method for getting the exact traveling wave solutions of the equation motion of single solitary wave is examined using the present methods the obtained results are indicated both in tabular and graphical form

global existence and nonexistence of solutions for second order nonlinear differential equations abstract this paper deals with the global existence and nonexistence of solutions of the second order nonlinear differential equation x x 0 satisfying x 0 x 0 and x 0 x 1 where is a positive parameter and with 0 and 0 is strictly increasing odd bijective and continuous on necessary and sufficient conditions are obtained for the initial value problem to have a unique global solution which is oscillatory and periodic examples are given to illustrate our main result finally a nonexistence result for the equation with a damping term is discussed as an application to our result

a petrov galerkin method for equal width equation abstract the equal width equation is solved numerically by petrov galerkin method using linear hat function and quadratic b spline function as trial and test functions respectively product approximation has been used in this method a linear stability analysis of the scheme is shown to be conditionally stable test problems including the single soliton and the interaction of solitons are used to validate the suggested method which is found to be accurate and efficient the maxwellian initial condition pulse and the development of an undular bore are also studied

on the electrostatic born infeld equation with extended charges in this paper we deal with the electrostatic born infeld equation left begin array ll operatorname div left displaystyle frac nabla phi sqrt 1 nabla phi 2 right rho quad in mathbb r n displaystyle lim x to infty phi x 0 end array right quad quad quad quad mathcal bi where rho is an assigned extended charge density we are interested in the existence and uniqueness of the potential phi and finiteness of the energy of the electrostatic field nabla phi we first relax the problem and treat it with the direct method of the calculus of variations for a broad class of charge densities assuming rho is radially distributed we recover the weak formulation of mathcal bi and the regularity of the solution of the poisson equation under the same smoothness assumptions in the case of a locally bounded charge we also recover the weak formulation without assuming any symmetry the solution is even classical if rho is smooth then we analyze the case where the density rho is a superposition of point charges and discuss the results in kiessling commun math phys 314 509 523 2012 other models are discussed as for instance a system arising from the coupling of the nonlinear klein gordon equation with the born infeld theory

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

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 complete classification of bifurcation diagrams for a class of p q laplace equations abstract we study the bifurcation problem of positive solutions for the one dimensional p q laplace equation with nonlinear term u r 1 there are five types of order relations for p q r we investigate the exact shape of the bifurcation curve in each type of the order relation we prove that there are two types of bifurcation curves that are increasing two types that are decreasing and one that is not monotone and turns exactly once moreover we study the asymptotic profile of the normalized solution u x u as u 0 or u where u denotes the l norm of u

remarks on time map for quasilinear equations abstract we present two different generalizations of r schaaf s 1990 18 time map formula to quasilinear equations including the case of p laplacian we give conditions for monotonicity and for convexity of the time map which imply uniqueness or multiplicity results for the corresponding dirichlet boundary value problem our time map formulas can be also used for effective computations of the global solution curves

a least squares finite element scheme for the ew equation abstract the equal width ew equation is solved by a least squares technique using linear space time finite elements in simulations of the migration of a single solitary wave this algorithm is shown to have good accuracy and conservation the development of solitary waves from an arbitrary initial condition within an ew environment is examined

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 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

a space time least square finite element scheme for advection diffusion equations abstract a space time least square finite element scheme is presented for the advection diffusion problems at moderate to high peclet numbers this scheme is designed to eliminate spurious oscillations and can be used to define the steady state solution as the asymptotic transient solution for large time numerical results using linear elements in a 1d space and bilinear elements in a 2d space demonstrate the accuracy and the stability of the new scheme

least squares finite element method for the advection diffusion equation the space time least squares finite element methods are constructed for the advection diffusion equation by using both linear shape functions and quadratic b spline shape functions two test problems are studied to demonstrate the accuracy of the present methods results of the two schemes have been compared

stochastic particle approximation for measure valued solutions of the 2d keller segel system we construct an approximation to the measure valued global in time solutions to the patlak keller segel model in 2d based on systems of stochastic interacting particles the advantage of our approach is that it reproduces the well known dichotomy in the qualitative behavior of the system and moreover captures the solution even after the possible blow up events we present a numerical method based on this approach and show some numerical results moreover we make a first step toward the convergence analysis of our scheme by proving the convergence of the stochastic particle approximation for the keller segel model with a regularized interaction potential the proof is based on a bbgky like approach for the corresponding particle distribution function

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

positive solutions to classes of infinite semipositone p q laplace problems with nonlinear boundary conditions abstract we consider one dimensional p q laplace problems u h t f u t 0 1 u 0 0 a u 1 g u 1 u 1 where 0 a 0 s s p 2 s s q 2 s 1 p q h c 0 1 0 f c 0 r with lim s 0 f s 0 and g c 0 0 0 such that g r s s is nondecreasing with respect to s 0 classifying the behaviors of f near infinity we establish the existence multiplicity and nonexistence of positive solutions in particular we provide a sufficient condition on f to obtain a multiplicity result for the case when lim s f s s r 1 0 1 r q which is new even in semilinear problems p q 2 the proofs are based on a krasnoselskii type fixed point theorem which is fit to overcome a lack of homogeneity

flocks herds and schools a quantitative theory of flocking we present a quantitative continuum theory of flocking the collective coherent motion of large numbers of self propelled organisms in agreement with everyday experience our model predicts the existence of an ordered phase of flocks in which all members of even an arbitrarily large flock move together with the same mean velocity stackrel ensuremath rightarrow v ensuremath ne 0 this coherent motion of the flock is an example of spontaneously broken symmetry no preferred direction for the motion is picked out a priori in the model rather each flock is allowed to and does spontaneously pick out some completely arbitrary direction to move in by analyzing our model we can make detailed quantitative predictions for the long distance long time behavior of this broken symmetry state the goldstone modes associated with this spontaneously broken rotational symmetry are fluctuations in the direction of motion of a large part of the flock away from the mean direction of motion of the flock as a whole these goldstone modes mix with modes associated with conservation of bird number to produce propagating sound modes these sound modes lead to enormous fluctuations of the density of the flock far larger at long wavelengths than those in e g an equilibrium gas our model is similar in many ways to the navier stokes equations for a simple compressible fluid in other ways it resembles a relaxational time dependent ginsburg landau theory for an n d component isotropic ferromagnet in spatial dimensions dg4 the long distance behavior is correctly described by a linearized theory and is equivalent to that of an unusual but nonetheless equilibrium model for spin systems for dl4 nonlinear fluctuation effects radically alter the long distance behavior making it different from that of any known equilibrium model in particular we find that in d 2 where we can calculate the scaling exponents exactly flocks exhibit a true long range ordered spontaneously broken symmetry state in contrast to equilibrium systems which cannot spontaneously break a continuous symmetry in d 2 the mermin wagner theorem we make detailed predictions for various correlation functions that could be measured either in simulations or by quantitative imaging of real flocks we also consider an anisotropic model in which the birds move preferentially in an easy e g horizontal plane and make analogous but quantitatively different predictions for that model as well for this anisotropic model we obtain exact scaling exponents for all spatial dimensions including the physically relevant case d 3

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

numerical schemes for a model for nonlinear dispersive waves abstract a description is given of a number of numerical schemes to solve an evolution equation that arises when modelling the propagation of water waves in a channel the discussion also includes the results of numerical experiments made with each of the schemes it is suggested on the basis of these experiments that one of the schemes may have discrete solitary wave solutions

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

a numerical solution of the equal width wave equation by a lumped galerkin method in this paper the equal width wave ew equation is solved by a numerical technique based on a lumped galerkin method using quadratic b spline finite elements to investigate the motion of a single solitary wave development of two solitary waves interaction and an undular bore the obtained results are compared with published numerical solutions a linear stability analysis of the method is also investigated

multiplicity of positive solutions to a p q laplacian equation involving critical nonlinearity abstract in this paper our main purpose is to establish the existence of multiple solutions of a p q laplacian equation with convex and sobolev critical nonlinearity by some standard variational methods whose key is to construct homotopies between and levels of the functional i and some analytical techniques

barriers of the mckean vlasov energy via a mountain pass theorem in the space of probability measures abstract we show that the empirical process associated with a system of weakly interacting diffusion processes exhibits a form of noise induced metastability the result is based on an analysis of the associated mckean vlasov free energy which for suitable attractive interaction potentials has at least two distinct global minimisers at the critical parameter value c on the torus one of these states is the spatially homogeneous constant state and the other is a clustered state we show that a third critical point exists at this value as a result we obtain that the probability of transition of the empirical process from the constant state scales like exp n with the energy gap at c the proof is based on a version of the mountain pass theorem for lower semicontinuous and geodesically convex functionals on the space of probability measures p 2 m equipped with the 2 wasserstein metric where m is a complete connected and smooth riemannian manifold

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

a complete classification of bifurcation diagrams of classes of multiparameter p laplacian boundary value problems we study the bifurcation diagrams of positive solutions of the multiparameter p laplacian problem p u x f u x 0 1 x 1 u 1 u 1 0 where p 1 p y y p 2y p u is the one dimensional p laplacian f u g u h u and 0 and 0 are two bifurcation parameters 0 and 0 are two given real numbers assuming that functions g and h satisfy hypotheses h1 h3 and h4a resp h1 h3 and h4b for fixed 0 resp 0 we give a classification of totally eight qualitatively different bifurcation diagrams we prove that on the u plane resp u plane each bifurcation diagram consists of exactly one curve which is either a monotone curve or has exactly one turning point where the curve turns to the right hence the problem has at most two positive solutions for each 0 resp 0 more precisely we prove the exact multiplicity of positive solutions in addition for all p 1 we give interesting examples which show complete evolution of bifurcation diagrams as resp varies

asymptotic length of bifurcation curves related to inverse bifurcation problems abstract we consider the nonlinear eigenvalue problem u t f u t 0 u t 0 t i 1 1 u 1 u 1 0 where f u u g u and 0 is a parameter our interest is the asymptotic length l g of the bifurcation curve g u 0 by the notion of l g we will propose a new framework of inverse bifurcation problems precisely we consider whether it is possible to characterize the unknown nonlinear term g u by l g if g u g 1 u u 2 sin u or g 2 u u 1 2 sin u then the bifurcation curves g i i 1 2 are continuous functions of 0 and g i 2 4 as furthermore they intersect the line 2 4 infinitely many times as in this paper by establishing the precise asymptotic formula for g i for 1 we obtain the precise asymptotic length l g i of g i for 1 then we consider the set g c 2 r g 2 4 as and study the existence and non existence of g whose asymptotic formula for l g is equal to those of l g i up to the second term

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

state transitions and the continuum limit for a 2d interacting self propelled particle system we study a class of swarming problems wherein particles evolve dynamically via pairwise interaction potentials and a velocity selection mechanism we find that the swarming system undergoes various changes of state as a function of the self propulsion and interaction potential parameters in this paper we utilize a procedure which connects a class of individual based models to their continuum formulations and determine criteria for the validity of the latter h stability of the interaction potential plays a fundamental role in determining both the validity of the continuum approximation and the nature of the aggregation state transitions we perform a linear stability analysis of the continuum model and compare the results to the simulations of the individual based one c 2007 elsevier b v all rights reserved

on a mean field optimal control problem abstract in this paper we consider a mean field optimal control problem with an aggregation diffusion constraint where agents interact through a potential in the presence of a gaussian noise term our analysis focuses on a pde system coupling a hamilton jacobi and a fokker planck equation describing the optimal control aspect of the problem and the evolution of the population of agents respectively the main contribution of the paper is a result on the existence of solutions for the aforementioned system we notice this model is in close connection with the theory of mean field games systems however a distinctive feature concerns the nonlocal character of the interaction it affects the drift term in the fokker planck equation as well as the hamiltonian of the system leading to new difficulties to be addressed