some upwinding techniques for finite element approximations of convection diffusion equations a uniform framework for the study of upwinding schemes is developed the standard finite element galerkin discretization is chosen as the reference discretization and differences between other discretization schemes and the reference are written as artificial diffusion terms these artificial diffusion terms are spanned by a four dimensional space of element diffusion matrices three basis matrices are symmetric rank one diffusion operators associated with the edges of the triangle the fourth basis matrix is skew symmetric and is associated with a rotation by 2 while finite volume discretizations may be written as upwinded galerkin methods the converse does not appear to be true our approach is used to examine several upwinding schemes including the streamline diffusion method the box method the scharfetter gummel discretization and a divergence free scheme

large time behavior of solutions for density suppressed motility system in higher dimensions abstract in this paper we study the asymptotic stability of the following density suppressed motility system u t v u u 1 u x t 0 v t v u v x t 0 in a bounded domain with smooth boundary where the motility function v and are positive under the conditions that the motility function v has the lower upper bounded and that is large enough we derive that system possesses a unique global solution in three dimensional space moreover we show that if the global classical solution exists of system in any dimensional space then the solution will converge to the equilibrium 1 1 exponentially as t when k 0 16 with k 0 max 0 v v 2 v

large time behavior of an attraction repulsion chemotaxis system abstract in this study we consider an attraction repulsion chemotaxis system u t u u v u w x t 0 v t v u v x t 0 w t w u w x t 0 with homogeneous neumann boundary conditions where and are positive parameters under the critical condition that 0 we prove that the system possesses a unique global solution which is uniformly bounded in the physical domain r n n 2 3 in addition we assert that we can find 0 0 such that for all of the initial data u 0 that satisfy u 0 0 the solution of the system approaches the steady state u 0 u 0 u 0 exponentially as time tends to infinity where u 0 1 u 0

a globally uniformly convergent finite element method for a singularly perturbed elliptic problem in two dimensions we analyze a new galerkin finite element method for numerically solving a linear convection dominated convection diffusion problem in two dimensions the method is shown to be convergent uniformly in the perturbation parameter of order h1 2 in a global energy norm which is stronger than the l2 norm this order is optimal in this norm for our choice of trial functions

a posteriori error estimates for upwind finite volume schemes for nonlinear conservation laws in multi dimensions in this paper we shall derive a posteriori error estimates in the l 1 norm for upwind finite volume schemes for the discretization of nonlinear conservation laws on unstructured grids in multi dimensions this result is mainly based on some fundamental a priori error estimates published in a recent paper by c chainais hillairet the theoretical results are confirmed by numerical experiments

global existence and asymptotic behavior of classical solutions to a fractional logistic keller segel system abstract in this paper we consider a fractional parabolic elliptic keller segel system with a logistic source on r n first we establish the regularity of weak solutions of the fractional parabolic equation using blow up arguments combined with liouville type theorems next by the semigroup method and regularity results we prove the local existence and uniqueness of classical solutions moreover the global existence and boundedness of classical solutions for given initial data are obtained under some conditions finally we show the asymptotic behavior of the global solutions with strictly positive initial data

aggregation vs global diffusive behavior in the higher dimensional keller segel model abstract we consider the classical parabolic parabolic keller segel system u t u u v x t 0 v t v v u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n it is proved that in space dimension n 3 for each q n 2 and p n one can find e 0 0 such that if the initial data u 0 v 0 satisfy u 0 l q e and v 0 l p e then the solution is global in time and bounded and asymptotically behaves like the solution of a discoupled system of linear parabolic equations in particular u v approaches the steady state m m as t where m is the total mass m u 0 of the population moreover we shall show that if is a ball then for arbitrary prescribed m 0 there exist unbounded solutions emanating from initial data u 0 v 0 having total mass u 0 m

how far does small chemotactic interaction perturb the fisher kpp dynamics abstract this paper deals with nonnegative solutions of the neumann initial boundary value problem for the fully parabolic chemotaxis growth system u e t u e e u e v e u e 1 u e x t 0 v e t v e v e u e x t 0 with positive small parameter e 0 in a bounded convex domain r n n 1 with smooth boundary the solutions converge to the solution u to the fisher kpp equation as e 0 it is shown that for all 0 and any suitably regular nonnegative initial data u i n i t v i n i t there are some constants e 0 0 and c 0 such that sup t 0 u e t u t l c e for all e 0 e 0

group actions on geodesic ptolemy spaces in this paper we study geodesic ptolemy metric spaces x which allow proper and cocompact isometric actions of crystallographic or more generally virtual polycyclic groups we show that x is equivariantly roughly isometric to a euclidean space

a new result for boundedness in the quasilinear parabolic parabolic keller segel model with logistic source abstract the current paper considers the boundedness of solutions to the following quasilinear keller segel model with logistic source ks u t d u u u v u u 2 x t 0 v t v u v x t 0 d u u u v v 0 x t 0 u x 0 u 0 x v x 0 v 0 x x where r n n 1 is a bounded domain with smooth boundary 0 and 0 one novelty of this paper is that we find a new a priori estimate u max 1 0 max 1 0 e x t d x so that we develop new l p estimate techniques and thereby obtain the boundedness results where c g n and 0 0 are the constants which are corresponding to the gagliardo nirenberg inequality see lemma 2 2 and the maximal sobolev regularity see lemma 2 3 to our best knowledge this seems to be the first rigorous mathematical result which indicates the relationship between m and that yields the boundedness of the solutions where m is the exponent of diffusion term d u the above mentioned results have significantly improved and extended previous results of several authors

convergence analysis for the full upwind finite volume solution of a convection diffusion problem abstract the paper is devoted to the study of convergence properties for an often used cell centered full upwind finite volume method fvm with voronoi boxes this fvm is applied to a convection diffusion problem the approach to proving convergence of the fvm is based on the construction of a nonconforming petrov galerkin finite element method fem such that the system of linear equations coincides completely with that of the fvm thus by proving convergence properties of the fem we obtain similar ones for the fvm for the error estimation of the fem the second strang lemma has to be modified

pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments abstract the current paper is concerned with pointwise persistence in full chemotaxis models with local as well as nonlocal time and space dependent logistic source in bounded domains we first prove the global existence and boundedness of nonnegative classical solutions under some conditions on the coefficients in the models next under the same conditions on the coefficients we show that pointwise persistence occurs that is any globally defined positive solution is bounded below by a positive constant independent of its initial condition when the time is large enough it should be pointed out that in 21 the authors established the persistence of mass for globally defined positive solutions which indicates that any extinction phenomenon if occurring at all necessarily must be spatially local in nature whereas the population as a whole always persists the pointwise persistence proved in the current paper implies that not only the population as a whole persists but also it persists at any location eventually it also implies the existence of strictly positive entire solutions

a flat strip theorem for ptolemaic spaces a metric space x d is called ptolemaic space pt space if the inequality xy zw xz yw xw yz holds for each quadruple of points x y z and w in x here xy d x y denotes the distance of two points x and y the main result proven in the paper is the following if x d is a pt space which is also a geodesic space i e for each pair x y there exists a geodesic connecting x and y and if x is moreover homeomorphic to 0 1 then x is isometric to a flat strip 0 a 2 with its euclidean metric the authors also give a new short proof of the fact that a proper geodesic pt space is always strictly distance convex see also th foertsch and v schroeder trans am math soc 363 no 6 2891 2906 2011 zbl 1220 53092 this also indicates a positive answer to the open question if every proper geodesic pt space is also a cat 0 space

boundedness of solutions in a chemotaxis system with nonlinear sensitivity and logistic source abstract this paper deals with a parabolic elliptic chemotaxis system with nonlinear sensitivity and logistic source u t u u v f u x t 0 0 v v g u x t 0 under homogeneous neumann boundary conditions in a smooth bounded domain r n n 1 where 0 the function u is the chemotactic sensitivity g u is the production rate of the chemoattractant and f u is the logistic source under some suitable assumptions on the nonlinearities u g u and logistic source f u we study the global boundedness of solutions for the problem

hyperbolicity cat 1 spaces and the ptolemy inequality using a four points inequality for the boundary of cat 1 spaces we study the relation between gromov hyperbolic spaces and cat 1 spaces

blowup and global solutions in a chemotaxis growth system abstract we study a keller segel type of system which includes growth and death of the chemotactic species and an elliptic equation for the chemo attractant the problem is considered in bounded domains with smooth boundary as well as in the whole space in case the random motion of the chemotactic species is neglected a hyperbolic elliptic problem results for which we characterize blow up of solutions in finite time and existence of regular solutions globally in time in dependence on the systems parameters in this case convexity of the domain is needed for the parabolic elliptic problem in dimensions three and higher we establish global existence of regular solutions in a limiting case which is an extension of the results given by tello and winkler 2007

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 existence and boundedness in a fully parabolic attraction repulsion chemotaxis system with signal dependent sensitivities and logistic source abstract this paper deals with the fully parabolic attraction repulsion chemotaxis system with signal dependent sensitivities and logistic source u t u u v v u w w u 1 u x t 0 v t v v u x t 0 w t w w u x t 0 under homogeneous neumann boundary conditions and initial conditions where r n n 2 is a bounded domain with smooth boundary are functions satisfying some conditions and 0 is a constant when are constants it is known that the above system possesses a globally bounded classical solution in some cases however there has been no work in the case that are functions this paper develops global existence and boundedness of classical solutions to the above system in such case

global existence and asymptotic behavior of solutions to a two species chemotaxis system with two chemicals in this paper we investigate the competitive parabolic elliptic parabolic elliptic two species chemotaxis system with two chemicals in a smooth bounded domain rn n 1 the existence of global bounded classical solution is proved under mild assumptions on the initial data and appropriate conditions on the strength of the damping death effects moreover for the case when both competition parameters a1 and a2 lie in 0 1 it is shown that such solution stabilizes to spatially homogeneous equilibria in the large time limit

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

a new approach toward stabilization in a two species chemotaxis model with logistic source abstract this paper aims at providing an alternative approach to study global dynamic properties for a two species chemotaxis model with the main novelty being that both populations mutually compete with the other on account of the lotka volterra dynamics more precisely we consider the following neumann initial boundary value problem u t d 1 u 1 u w 1 u 1 u a 1 v x t 0 v t d 2 v 2 v w 2 v 1 a 2 u v x t 0 0 d 3 w w u v x t 0 in a bounded domain r n n 1 with smooth boundary where d 1 d 2 d 3 1 2 1 2 a 1 a 2 are positive constants when a 1 0 1 and a 2 0 1 it is shown that under some explicit largeness assumptions on the logistic growth coefficients 1 and 2 the corresponding neumann initial boundary value problem possesses a unique global bounded solution which moreover approaches a unique positive homogeneous steady state u v w of above system in the large time limit the respective decay rate of this convergence is shown to be exponential when a 1 1 and a 2 0 1 if 2 is suitable large for all sufficiently regular nonnegative initial data u 0 and v 0 with u 0 0 and v 0 0 the globally bounded solution of above system will stabilize toward 0 1 1 as t in algebraic

piecewise linear petrov galerkin error estimates for the box method the perpendicular bisector box method pbm discretization of the laplacian on a simplicial element is equivalent to a piecewise linear petrov galerkin method with piecewise constant test functions this pbm element matrix is expressed in terms of galerkin s piecewise linear plg element matrix and the average values assumed by test functions over element faces only these results are shown to be applicable to the accuracy analysis of pbm discretizations of extremum stable elliptic gradient equations subject to regularity conditions on variable coefficients in these equations these gradient equations are shown to satisfy sufficient conditions to validate a discrete extremum principle on n dimensional delaunay triangulations this conclusion need not hold for the plg equations the dependence of the discretized laplacian on only the element face averages of test functions enables the substitution of piecewise polynomial test functions for the piecewise constant functions the latter are used to establish piecewise linear accuracy for the pbm by an adaptation of an error estimate for petrov galerkin methods by strang

sub logistic source can prevent blow up in the 2d minimal keller segel chemotaxis system it is well known that the neumann initial boundary value problem for the minimal keller segel chemotaxis system in a 2d bounded smooth domain has no blow ups for any presence of logistic source of cell kinetics here for a large class of cell kinetics including sub logistic sources we find an explicit condition involving the chemotactic strength the asymptotic damping rate and the initial mass of cells to ensure the uniform in time boundedness for the corresponding 2d neumann initial boundary value problem our finding in particular shows that sub logistic source can prevent blow up in 2d indicating that logistic damping is not the weakest damping to guarantee boundedness for the 2d keller segel minimal chemotaxis model

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

large time behavior of solution to a fully parabolic chemotaxis haptotaxis model in higher dimensions abstract this paper deals with 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 n with zero flux boundary conditions where and are positive parameters it is shown that if is suitably large then for all sufficiently smooth initial data the associated initial boundary value problem possesses a unique global in time classical solution that is bounded in 0 and if the initial data w 0 is small w becomes asymptotically negligible moreover we prove that when domain is convex 1 1 0 is globally asymptotically stable provided that u 0 0 and thereby extends the result of hillen et al 2013 18 to the higher space dimensions

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

stabilization in the logarithmic keller segel system abstract the keller segel system u t d u d u v v x t 0 v t d v v u x t 0 is considered in a bounded domain r n n 2 with smooth boundary where 0 and d 0 the main results identify a condition on the parameters 2 n and d 0 essentially reducing to the assumption that 2 d be suitably small under which for all reasonably regular and positive initial data the corresponding classical solution of an associated neumann initial boundary value problem known to exist globally according to previous findings approaches the homogeneous steady state u 0 u 0 at an exponential rate with respect to the norm in l 2 as t where u 0 1 u 0 as a particular consequence this entails a convergence statement of the above flavor in the normalized system with d 1 and fixed 2 n provided that satisfies a certain smallness condition

a brief survey on numerical methods for solving singularly perturbed problems in the present paper a brief survey on computational techniques for the different classes of singularly perturbed problems is given this survey is a continuation of work performed earlier by the first author and contains the literature of the work done by the researchers during the years 2000 2009 however some older important relevant papers are also included in this survey we also mentioned those papers which are not surveyed in the previous survey papers by the first author of this paper see appl math comput 30 1989 223 259 130 2002 457 510 134 2003 371 429 for details thus this survey paper contains a surprisingly large amount of literature on singularly perturbed problems and indeed can serve as an introduction to some of the ideas and methods for the singular perturbation problems

asymptotic behavior of solutions to two dimensional chemotaxis system with logistic source and singular sensitivity abstract in this paper we study the asymptotic behavior of solutions to the parabolic elliptic chemotaxis system with singular sensitivity and logistic source u t u u v v r u u 2 0 v v u subject to homogeneous neumann boundary conditions in a bounded domain r 2 with smooth boundary where 0 and r r it is proved that for any nonnegative initial date u 0 c such that u 0 1 16 2 2 with the constant relying on the solution u t v t converges asymptotically to the constant equilibrium r r in the l norm as t if r 2 1 1 2 2 16

on convergence to equilibria for the keller segel chemotaxis model we show that any global in time bounded solution to the keller segel chemotaxis model converges to a single equilibrium as time tends to infinity the proof is based on a generalized version of the lojasiewicz simon theorem

eventual smoothness and asymptotics in a three dimensional chemotaxis system with logistic source abstract we prove existence of global weak solutions to the chemotaxis system u t u u v u u 2 v t v v u under homogeneous neumann boundary conditions in a smooth bounded convex domain r n for arbitrarily small values of 0 additionally we show that in the three dimensional setting after some time these solutions become classical solutions provided that is not too large in this case we also consider their large time behaviour we prove decay if 0 and the existence of an absorbing set if 0 is sufficiently small

global existence and asymptotic behavior of classical solutions to a parabolic elliptic chemotaxis system with logistic source on rn in the current paper we consider the following parabolic elliptic semilinear keller segel model on rnrn ut u u v au bu2 x rn t 0 0 i v u x rn t 0 where 0 a 0 b 0 are constant real numbers and n is a positive integer we first prove the local existence and uniqueness of classical solutions u x t u0 v x t u0 u x t u0 v x t u0 with u x 0 u0 u0 x u x 0 u0 u0 x for various initial functions u0 x u0 x next under some conditions on the constants a b a b and the dimension n we prove the global existence and boundedness of classical solutions u x t u0 v x t u0 u x t u0 v x t u0 for given initial functions u0 x u0 x finally we investigate the asymptotic behavior of the global solutions with strictly positive initial functions or nonnegative compactly supported initial functions under some conditions on the constants a b a b and the dimension n we show that for every strictly positive initial function u0 u0 limt supx rn u x t u0 ab v x t u0 ab 0 and that for every nonnegative initial function u0 u0 with non empty and compact support supp u0 supp u0 there are 0 clow u0 cup u0 such that limt sup x ct u x t u0 ab v x t u0 ab 0 0 c clow u0 and limt sup x ct u x t u0 v x t u0 0 c cup u0

large time behavior of solutions to a chemotaxis model with porous medium diffusion abstract in this paper we study the large time behavior of a chemotaxis model with nonlinear diffusion and consumption u t u m u v u 1 u v t v v u where m 1 in a previous paper 5 we have proved the existence and uniform boundedness of global weak solutions for any nonnegative initial data and any m 1 in this work we show that the weak solutions strongly converge to 1 0 in the large time limit

convergence rate estimates of solutions in a higher dimensional chemotaxis system with logistic source abstract we study the global attractors to the chemotaxis system with logistic source u t u u v a u b u 2 v t v v u in r subject to the homogeneous neumann boundary conditions where smooth bounded domain r n with b 0 a r and 0 1 for the parabolic elliptic case with 0 and n 3 we obtain that the positive constant equilibrium a b a b is a global attractor if a 0 and b max n 2 n a 4 under the assumption n 3 it is proved that for either the parabolic elliptic case with 0 a 0 b max 3 a 4 or the parabolic parabolic case with 1 a 0 b a 4 large enough the system admits the positive constant equilibrium a b a b as a global attractor while the trivial equilibrium 0 0 is a global attractor if a 0 and b 0 it is pointed out that here the convergence rates are established for all of them the results of the paper mainly rely on parabolic regularity theory and lyapunov functionals carefully constructed

convergence rate of a quasilinear parabolic elliptic chemotaxis system with logistic source abstract this paper deals with the quasilinear parabolic elliptic chemotaxis system u t d u u u v u 1 u x t 0 0 v v u x t 0 under homogeneous neumann boundary conditions in a bounded domain r n with smooth boundary where 0 and 0 d u is supposed to satisfy d 0 0 d u u with 0 1 when n 2 the convergence rate of the solution is investigated

analysis of a cell vertex finite volume method for convection diffusion problems a cell vertex finite volume approximation of elliptic convection dominated diffusion equations is considered in two dimensions the scheme is shown to be stable and second order convergent in a mesh dependent l 2 norm

boundedness of solutions to a quasilinear parabolic elliptic keller segel system with logistic source abstract we study global solutions of a class of chemotaxis systems generalizing the prototype u t u 1 m 1 u u u 1 q 1 v a u b u r x t 0 0 v v u x t 0 in a bounded domain r n n 1 with smooth boundary with parameters m 1 r 1 a 0 b q 0 it is shown that when q 1 max r m 2 n or b b 0 n r m 2 r m n 2 r 2 if q 1 r then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded the results improve the results of wang et al 2014 37 and cao and zheng 2014 6

global bounded solutions and their asymptotic properties under small initial data condition in a two dimensional chemotaxis system for two species abstract in this paper a fully parabolic chemotaxis system for two species u t u u w x t 0 v t v v w x t 0 w t w u w v w x t 0 is considered under homogeneous neumann boundary conditions in a smooth bounded domain r 2 we obtain the global boundedness and asymptotic behavior with small initial data condition in critical space more precisely it is proved that one can find a small e 0 0 such that for any initial data u 0 v 0 w 0 satisfying u 0 l 1 e 0 and w 0 l 2 e 0 the solution of the problem above is global in time and bounded in addition u v w converges to the steady state m 1 m 2 m 1 1 m 2 as t where m 1 1 u 0 and m 2 1 v 0

boundedness of solutions to a quasilinear parabolic parabolic keller segel system with a logistic source abstract we study global solutions of a class of chemotaxis systems generalizing the prototype u t u u u v a u b u r x t 0 v t v v u x t 0 in a bounded domain r n n 1 with smooth boundary u u 1 u u u 1 1 and the parameters r 1 a 0 b 0 and r it is proved that if 0 max r 1 2 n or b is big enough if r 1 then the classical solutions to the above system are uniformly in time bounded our results improve the results of wang et al 2014 28 and cao 2014 3 and also enlarge the range of the results of tao and winkler 2012 25 and ishida et al 2014 14

global existence and boundedness in a chemotaxis haptotaxis system with signal dependent sensitivity abstract this paper deals with the chemotaxis haptotaxis system with signal dependent sensitivity u t u v 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 under homogeneous neumann boundary conditions and initial conditions where r n n 3 is a bounded domain with smooth boundary 0 are constants and is a function satisfying some conditions in the case that is a constant it is known that the above system possesses a global classical solution under some conditions cao 4 tao 10 tao and winkler 11 however in the case that is a function the above system has not been studied the purpose of this paper is to establish global existence and boundedness in the above system