existence of weak solutions for a hyperbolic model of chemosensitive movement a hyperbolic model for chemotaxis and chemosensitive movement in one space dimension is considered in contrast to parabolic models for chemotaxis the hyperbolic model allows us to take the dependence of the particle speed on external stimuli explicitly into account this qualitatively covers recent experiments on chemotaxis in which it has been shown that particles adapt their speed to the surrounding environment the model presented here consists of two hyperbolic differential equations of first order coupled with an elliptic equation we assume that the speed depends on the external stimulus only and not on its gradients in that case solutions with steep gradients are expected which have the interpretation of moving swarms a notion of weak solutions for this hyperbolic chemotaxis model is presented and the global existence of weak solutions is shown the proof relies on the vanishing viscosity method i e we obtain the weak solution as the limit of classical solutions of an associated parabolically regularized problem for vanishing viscosity parameter numerical simulations demonstrate phenomena like swarming behaviour and formation of steep gradients

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

the drift diffusion system in two dimensional critical hardy space abstract we establish the time local well posedness for large data of a solution to two dimensional drift diffusion system in the critical hardy space h 1 r 2 this solution is obtained in the hardy class for which the natural free energy is well defined from the initial time the time global well posedness result for small data is also obtained in a similar class the crucial parts of the proof are to employ the end point type of maximal regularity for the homogeneous heat equation and some new bilinear estimates in the hardy space

finite time blow up for a solution to system of the drift diffusion equations in higher dimensions we discuss the existence of a blow up solution for a multi component parabolic elliptic drift diffusion model in higher space dimensions we show that the local existence uniqueness and well posedness of a solution in the weighted l 2 spaces moreover we prove that if the initial data satisfies certain conditions then the corresponding solution blows up in a finite time this is a system case for the blow up result of the chemotactic and drift diffusion equation proved by nagai j inequal appl 6 37 55 2001 and nagai et al hiroshima j math 30 463 497 2000 and gravitational interaction of particles by biler colloq math 68 229 239 1995 biler and nadzieja colloq math 66 319 334 1994 adv differ equ 3 177 197 1998 we generalize the result in kurokiba and ogawa differ integral equ 16 427 452 2003 differ integral equ 28 441 472 2015 and kurokiba differ integral equ 27 5 6 425 446 2014 for the multi component problem and give a sufficient condition for the finite time blow up of the solution the condition is different from the one obtained by corrias et al milan j math 72 1 28 2004

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

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

numerical simulation of nonlinear continuity equations by evolving diffeomorphisms abstract in this paper we present a numerical scheme for nonlinear continuity equations which is based on the gradient flow formulation of an energy functional with respect to the quadratic transportation distance it can be applied to a large class of nonlinear continuity equations whose dynamics are driven by internal energies given external potentials and or interaction energies the solver is based on its variational formulation as a gradient flow with respect to the wasserstein distance positivity of solutions as well as energy decrease of the semi discrete scheme are guaranteed by its construction we illustrate this property with various examples in spatial dimension one and two

quiver varieties and lusztig s algebra abstract we study preprojective algebras of graphs and their relationship to module categories over representations of quantum sl 2 as an application ade quiver varieties of nakajima are shown to be subvarieties of the variety of representations of a certain associative algebra introduced by lusztig

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

numerical and experimental investigations of the nonlinear dynamics and chaos in non smooth systems nature is full of non smooth nonlinearities that are usually related to either friction phenomena or the discontinuous characteristics of intermittent contacts in general non smooth characteristics cause difficulties in the modeling and simulation of natural systems a smoothed switch model is used in this contribution in order to analyze non smooth systems the procedure is shown to be an effective method of dealing with this kind of system and has advantages in numerical implementation as an application of the general formulation a single degree of freedom oscillator with discontinuous support is analyzed an experimental apparatus is developed in order to verify the capability of the mathematical formulation and the numerical procedure used to describe the system general behavior this apparatus is instrumented to obtain all system state variables making it possible to compare experimental results with those obtained through numerical simulations the system s dynamical behavior shows a rich response that includes dynamical jumps bifurcations and chaos different aspects related to nonlinear dynamics are carried out showing that numerical simulations are in close agreement with experimental data

fredholmness and smooth dependence for linear time periodic hyperbolic systems abstract this paper concerns n n linear one dimensional hyperbolic systems of the type t u j a j x x u j k 1 n b j k x u k f j x t j 1 n with periodicity conditions in time and reflection boundary conditions in space we state conditions on the data a j and b j k and the reflection coefficients such that the system is fredholm solvable moreover we state conditions on the data such that for any right hand side there exists exactly one solution that the solution survives under small perturbations of the data and that the corresponding data to solution map is smooth with respect to appropriate function space norms in particular those conditions imply that no small denominator effects occur we show that perturbations of the coefficients a j lead to essentially different results than perturbations of the coefficients b j k in general our results cover cases of non strictly hyperbolic systems as well as systems with discontinuous coefficients a j and b j k but they are new even in the case of strict hyperbolicity and of smooth coefficients

mesenchymal motion models in one dimension mesenchymal motion denotes a form of cell movement through tissue which can be observed for certain cancer metastases in t hillen j math biol 53 2006 pp 585 616 a mathematical model for this form of movement was introduced in the current paper we present a comprehensive analysis of the one dimensional mesenchymal motion model we establish the global existence of classical solutions and rigorously carry out the parabolic limit of the model we discuss the stationary solutions prove the existence of traveling wave solutions and use numerical simulations to illustrate the results finally we discuss the biological implications of the results

computing chevalley bases in small characteristics abstract let l be the lie algebra of a simple algebraic group defined over a field f and let h be a split maximal toral subalgebra of l then l has a chevalley basis with respect to h if char f 2 3 it is known how to find it in this paper we treat the remaining two characteristics to this end we present a few new methods implemented in magma which vary from the computation of centralizers of one root space in another to the computation of a specific part of the lie algebra of derivations of l

regularized friction and continuation comparison with coulomb s law abstract periodic solutions of systems with friction are difficult to investigate because of the non smooth nature of friction laws this paper examines periodic solutions and most notably stick slip on a simple one degree of freedom system mass spring damper and belt with coulomb s friction law and with a regularized friction law i e the friction coefficient becomes a function of relative speed with a stiffness parameter with coulomb s law the stick slip solution is constructed step by step which gives a usable existence condition with the regularized law the asymptotic numerical method and the harmonic balance method provide bifurcation diagrams with respect to the belt speed or normal force and for several values of the regularization parameter formulations from the coulomb case give the means of a comparison between regularized solutions and a standard reference with an appropriate definition regularized stick slip motion exists its amplitude increases with respect to the belt speed and its pulsation decreases with respect to the normal force

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

global well posedness and blow up for the 2 d patlak keller segel equation abstract in this paper we prove that for every nonnegative initial data in l 1 r 2 the patlak keller segel equation is globally well posed if and only if the total mass m 8 our proof is based on some monotonicity formulas of nonnegative mild solutions

preventing blow up in a chemotaxis model abstract this paper deals with a chemotactic model of keller segel type the main feature of the keller segel model is the possibility of blow up of solutions in a finite time to eliminate the possibility of blow up a modified version of keller segel model is introduced the blow up control relies on the presence of a pressure function which increases faster than a logarithm for high enough cells densities for such a pressure function the solutions cannot blow up in a finite time some other conditions are introduced to ensure the global boundedness

stability of piecewise linear oscillators with viscous and dry friction damping abstract the stability analysis for periodic motions of a class of harmonically excited single degree of freedom oscillators with piecewise linear characteristics is presented the common characteristic of these oscillators is that they posses viscous and constant damping properties which depend on their velocity direction the presence of constant damping terms in the equation of motion introduces acceleration discontinuities and makes possible the appearance of finite time intervals within the periodic solution where the oscillator is stuck at the same position harmonic and subharmonic motions with an arbitrary number of solution pieces are examined the analysis takes advantage of the fact that the exact solution form for any solution piece included between two consecutive zero velocity values is known it is based on the derivation of a matrix relation which determines how an arbitrary but small perturbation at the beginning of a periodic solution propagates to the end of a response period then results obtained by bifurcation analysis of the periodic solutions are also presented at the end some of the analytical predictions are confirmed by considering an example mechanical model

exponential dichotomy for hyperbolic systems with periodic boundary conditions abstract we investigate evolution families generated by general linear first order hyperbolic systems in one space dimension with periodic boundary conditions we state explicit conditions on the coefficient functions that are sufficient for the existence of exponential dichotomies on r in the space of continuous periodic functions

point dynamics in a singular limit of the keller segel model 2 formation of the concentration regions this paper continues the analysis started in the first part of this article cf j j l velazquez siam j appl math 64 2004 pp 1198 1223 it was seen there using the method of matched asymptotics that a regularized version of the keller segel system admits for a suitable asymptotic limit solutions with some regions of high concentrations for the cell density this paper considers the relation between the phenomenon of blow up for the limit problem and the dynamics of the concentration regions described in j j l velazquez siam j appl math 64 2004 pp 1198 1223 in particular this paper analyzes the precise way in which the regularization introduced in the keller segel system stops the aggregation process and yields the formation of concentration regions

inverses of cartan matrices of lie algebras and lie superalgebras abstract we derive explicit formulas for the inverses of the cartan matrices of the simple lie algebras and the basic classical lie superalgebras as well as for their infinite generalizations

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

dynamics of friction oscillators excited by a moving base and or driving force abstract the response of a single degree of freedom system with dry friction under a constant velocity of the base and or harmonic driving force is analyzed via either closed form or numerical approaches the first main purpose of this paper is to investigate the influence of the base speed on the system response to this end in the case of a moving base with and without a harmonic driving force as a new result closed form solutions are presented under the assumption of coulomb s friction law including a static coefficient different from the kinetic one in more detail the existence of a critical base velocity is proved which is the lower bound of no stick and base velocity independent motions the second main purpose of this paper is to investigate the influence of friction modelling on the system response to this end the results achieved via the coulomb s law are compared with those obtained via a particular velocity dependent friction law the proposed law allows the static coefficient to exponentially decay to the kinetic one still preserving the discontinuity at null relative speed the purpose at hand has also been accomplished by using standard numerical methods

self similar solutions to a parabolic system modeling chemotaxis we study the forward self similar solutions to a parabolic system modeling chemotaxisut u u v vt v u in the whole space r2 where is a positive constant using the liouville type result and the method of moving planes it is proved that self similar solutions u v must be radially symmetric about the origin then the structure of the set of self similar solutions is investigated as a consequence it is shown that there exists a threshold in r2u for the existence of self similar solutions in particular for 0 1 2 there exists a self similar solution u v if and only if r2u 8

fredholm solvability of time periodic boundary value hyperbolic problems abstract we investigate a large class of linear boundary value problems for the general first order one dimensional hyperbolic systems in the strip 0 1 r we state rather broad natural conditions on the data under which the operators of the problems satisfy the fredholm alternative in the spaces of continuous and time periodic functions a crucial ingredient of our analysis is a non resonance condition which is formulated in terms of the data responsible for the bijective part of the fredholm operator in the case of 2 2 systems with reflection boundary conditions we provide a criterium for the non resonant behavior of the system

stability of some mechanisms of chemotactic aggregation in this paper the stability of a blow up mechanism for a particular keller segel model that yields chemotactic aggregation is considered it is shown using matched asymptotic methods that the considered blow up mechanism is stable more precisely small perturbations of the initial data produce a small shifting on the blow up time and the blow up point but not modification in the blow up mechanism

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

invariance principles for hyperbolic random walk systems abstract reaction random walk systems are hyperbolic models for the description of spatial motion in one dimension and reaction of particles in contrast to reaction diffusion equations particles have finite propagation speed for parabolic systems invariance results and maximum principles are well known a convex set is positively invariant if at each boundary point an outer normal is a left eigenvector of the diffusion matrix and if the vector field defined by the pure reaction equation points inward at the boundary here we show a corresponding result for random walk systems the model parameters are the particle speeds the rates of change in direction and the reaction vector field a convex domain is invariant if at each boundary point an outer normal is a left eigenvector of the speed matrix and if a vector field given by the reaction equation combined with the turning rates points inward finally a positivity result is shown

a branching random evolution and a nonlinear hyperbolic equation particles moving at constant speed along the real line and reversing direction as a poisson process constitute a random evolution if the particles simultaneously undergo an independent branching p

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

asymptotic preserving and well balanced schemes for the 1d cattaneo model of chemotaxis movement in both hyperbolic and diffusive regimes abstract the original well balanced wb framework of greenberg and leroux 1996 24 and gosse 2002 18 relying on non conservative nc products see lefoch and tzavaras 1999 41 is set up in order to efficiently treat the so called cattaneo model of chemotaxis in 1d see hillen and stevens 2000 29 it proceeds by concentrating the source terms onto dirac masses this allows to handle them by nc jump relations based on steady state equations which can be integrated explicitly a riemann solver is deduced and the corresponding wb godunov scheme completed with the standard hoff smoller theory see hillen and stevens 2000 29 for the diffusion reaction equation ruling the evolution of the chemoattractant concentration is studied in detail later following former results of gosse and toscani 2002 2003 21 22 a simple rewriting of the nc jump relations allows to generate another version of the same godunov scheme which is well adapted to the parabolic scaling involving a small parameter e the standard bv framework is used to study the uniform stability of this asymptotic preserving ap scheme with respect to e allows to pass to the limit and derive a simple centered discretization of the keller segel model finally results by filbet 2006 14 permit to pass to the complementary limit when the space step h is sent to zero numerical results are included to illustrate the feasibility and the efficiency of the method

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

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

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

global boundedness in a three dimensional chemotaxis haptotaxis model abstract this paper studies the chemotaxis haptotaxis model of cancer invasion u t u u v u w u 1 u w x t 0 v t v v u x t 0 w t v w x t 0 in a bounded smooth domain r 3 with zero flux boundary conditions where the parameters and are positive it is shown that the corresponding initial boundary problem possesses a unique classical solution which is global in time and bounded under the explicit condition 11 2 2 w 0 l 2 2 37 2 4 2 w 0 l 2 and suitable regularity assumptions on the initial data u 0 v 0 w 0

global boundedness in a quasilinear attraction repulsion chemotaxis model with nonlinear sensitivity abstract this paper considers the attraction repulsion chemotaxis model with homogeneous neumann boundary conditions in a smooth bounded convex domain in this model when the scaling constant is zero and the chemotactic sensitivity functions are nonlinear we prove that this system possesses a unique global classical solution that is uniformly bounded under some assumptions when the scaling constant is one and one of the chemotactic sensitivity functions is nonlinear we also obtain a unique bounded global classical solution under some assumptions

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

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

well posedness for chemotaxis dynamics with nonlinear cell diffusion abstract this paper investigates the well posedness of a reaction diffusion system of chemotaxis type with a nonlinear diffusion coefficient and a dynamics growth death of the cell population b and a stationary equation for the chemoattractant c with respect to other works in which a nonlinear diffusion for cells has been considered we treat here two distinct cases for this diffusion coefficient the first in which it is a positive bounded function on r and the other in which it may display a singularity at a finite value of the cell density essentially the latter model is new and describes the saturation of the cell population in the neighborhood of a critical value for its diffusion coefficient the chemotactic sensitivity function is supposed to depend both on the cell and chemoattractant densities for homogeneous neumann boundary conditions for the cell population and chemoattractant we prove the existence of a local in time solution when the l 2 norm of the initial datum b 0 is sufficiently small and compute the maximum time interval for which the solution is bounded and smooth under a stronger assumption related to the chemotactic sensitivity we show that there exists a global in time solution for arbitrarily large initial data in a case when the diffusion coefficient is singular we focus on a model expressed by a variational inequality describing the saturation of the cell population at the blowing up diffusion value here the proof requires the study of an intermediate problem with robin boundary conditions which may be interesting by itself in all situations uniqueness follows on a time interval included not necessarily strictly in that of the solution existence under sufficient conditions

blow up behavior of solutions to a degenerate parabolic parabolic keller segel system qualitative properties of solutions blowing up in finite time are obtained for a degenerate parabolic parabolic keller segel system the nonlinear diffusion being of porous medium type with an exponent smaller or equal to the critical one m c 2 n 1 n in both cases it is shown that only type ii blow up is possible that is blow up at the same rate as backward self similar solutions never occurs further information on the generation of singularities induced by mass concentration are given in the critical case

interplay of chemotaxis and chemokinesis mechanisms in bacterial dynamics motivated by observations of the dynamics of myxococcus xanthus we present a self interacting random walk model that describes the competition between chemokinesis and chemotaxis cells are constrained to move in one dimension but release a chemical chemoattractant at a steady state the bacteria sense the chemical that they produce the probability of direction reversals is modeled as a function of both the absolute level of chemoattractant sensed directly under each cell as well as the gradient sensed across the length of the cell if the chemical does not degrade or diffuse rapidly the one dimensional trajectory depends on the entire past history of the trajectory we derive the corresponding fokker planck equations use an iterative mean field approach that we solve numerically for short times and perform extensive monte carlo simulations of the model cell positional distributions and the associated moments are computed in this feedback system average drift and mean squared displacements are found crossover behaviors among different diffusion regimes are found

global existence of solutions and uniform persistence of a diffusive predator prey model with prey taxis this paper proves the global existence and boundedness of solutions to a general reaction diffusion predator prey system with prey taxis defined on a smooth bounded domain with no flux boundary condition the result holds for domains in arbitrary spatial dimension and small prey taxis sensitivity coefficient this paper also proves the existence of a global attractor and the uniform persistence of the system under some additional conditions applications to models from ecology and chemotaxis are discussed

asymptotic behavior of equilibrium states of reaction diffusion systems with mass conservation abstract we deal with a stationary problem of a reaction diffusion system with a conservation law under the neumann boundary condition it is shown that the stationary problem turns to be the euler lagrange equation of an energy functional with a mass constraint when the domain is the finite interval 0 1 we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d the ratio of the diffusion coefficient of the system tends to zero in view of a logarithmic function in the leading term of the potential we get to a scaling parameter satisfying the relation e d log 2 the main result shows that a sequence of minimizers converges to a dirac mass multiplied by the total mass and that by a scaling with the asymptotic profile exhibits a parabola in the nonvanishing region we also prove the existence of an unstable monotone solution when the mass is small

global and exploding solutions for nonlocal quadratic evolution problems nonlinear nonlocal parabolic equations modeling the evolution of density of mutually interacting particles are considered the inertial type nonlinearity is quadratic and nonlocal while the diffusive term also nonlocal is anomalous and fractal i e represented by a fractional power of the laplacian conditions for global in time existence versus finite time blow up are studied self similar solutions are constructed for certain homogeneous initial data

existence and uniqueness theorem on mild solutions to the keller segel system in the scaling invariant space we prove the existence and uniqueness of solutions u v to the keller segel system of parabolic parabolic type in rn for n 3 in the scaling invariant class u lq 0 t lr rn v lr 0 t h r rn where 2 q n r 2 2 q n r 2 provided the initial data u0 v0 is chosen as u0 ln 2 rn v0 h 2 n 2 rn for n 2 n 2 1 2 in particular our uniqueness result holds for all n 2 even though we impose an assumption only on u such as lq 0 t lr rn for 2 q n r 2 with n 2 r as for the marginal case when r n 2 we show that if n 3 and if u c 0 t ln 2 rn v c 0 t ln rn then u v is the only solution

on the stability of critical chemotactic aggregation we consider the two dimensional parabolic elliptic keller segel model of chemotactic aggregation for radially symmetric initial data we show the existence of a stable mechanism of singularity formation and obtain a complete description of the associated aggregation process

type ii blow up mechanisms in a semilinear heat equation with critical joseph lundgren exponent abstract we are concerned with blow up mechanisms in a semilinear heat equation u t u u p 1 u x r n t 0 where p 1 is a constant it is well known that type ii blow up does occur if n 11 and p p jl where p jl stands for the joseph lundgren exponent p jl n 10 1 4 n 4 2 n 1 n 11 on the other hand it is also known that if p p jl type ii blow up cannot occur under mild assumptions on initial data as far as nonnegative radially symmetric solutions are concerned it has long remained open whether or not type ii blow up occurs for the borderline case p p jl n 11 in the present article we prove constructively the existence of type ii blow up solutions for the joseph lundgren critical case thus giving an answer to this question

inverses of cartan matrices of lie algebras and lie superalgebras abstract the inverses of indecomposable cartan matrices are computed for finite dimensional lie algebras and lie superalgebras over fields of any characteristic and for hyperbolic almost affine complex lie super algebras this enables one to express the fundamental weights in terms of simple roots corresponding to the cartan matrix we discovered three yet inexplicable new phenomena of which a and b concern hyperbolic almost affine complex lie super algebras a several of the inverses of cartan matrices have all their elements negative not just non positive as they should be according to an a priori characterization due to zhang hechun b the 0s only occur on the main diagonals of the inverses c the determinants of inequivalent cartan matrices of the simple lie super algebra of any dimension may differ in any characteristic we interpret most of the results of wei and zou 2017 31 as inverses of the gram matrices of non degenerate invariant symmetric bilinear forms on the super algebras considered not of cartan matrices and give more adequate references in particular the inverses of cartan matrices of simple lie algebras were already published starting with dynkin s paper in 1952 see also table 2 in springer s book by onishchik and vinberg 1990

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

exponential speed of uniform convergence of the cell density toward equilibrium for subcritical mass in a patlak keller segel model this paper is concerned with a chemotaxis aggregation model for cells more precisely with a parabolic elliptic semilinear patlak keller segel system in a ball of rn exhibiting a critical mass phenomenon for n 2 the main result of this paper is the exponential speed of uniform convergence of radial solutions toward the unique steady state in the subcritical case we stress that this covers in particular the classical keller segel system with n 2 well known for its critical mass 8 and that the result improves on the known results even for this most studied problem a key tool is an associated one dimensional degenerate parabolic problem the proof exploits its formal gradient flow structure ut f u t on an infinite dimensional riemannian manifold in particular we show a new hardy type inequality equivalent to the strict convexity of f at any steady state of subcritical mass which heuristically explains the exponential speed of convergence

a sufficient condition for type i blowup in a parabolic elliptic system abstract this paper is concerned with blowup of positive solutions to a cauchy problem for a parabolic elliptic system u t u u v in r n 0 t 0 v u in r n 0 t we say that a solution u v blows up at t t if lim sup t t u t with l norm in r n when a solution u v blows up at t t the blowup is called of type i if lim sup t t t t u t and of type ii otherwise it was shown in herrero and velazquez 1996 12 and mizoguchi and senba 2007 20 that there exist radial type ii blowup solutions for n 2 or n 11 when 3 n 9 type ii blowup solutions were given by formal analysis in brenner et al 1999 2 and herrero et al 1997 10 we give a sufficient condition for a solution to exhibit type i blowup in the case of 3 n 9